https://math.byu.edu/wiki/api.php?action=feedcontributions&user=Roundy&feedformat=atomMathWiki - User contributions [en]2021-01-23T08:03:53ZUser contributionsMediaWiki 1.26.3https://math.byu.edu/wiki/index.php?title=Math_674:_Lie_Groups_and_Algebras&diff=2366Math 674: Lie Groups and Algebras2015-01-07T17:28:29Z<p>Roundy: /* Minimal learning outcomes */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Lie Groups and Algebras<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Offered ===<br />
Upon request.<br />
<br />
=== Prerequisite ===<br />
673, equivalent, or teacher approval.<br />
<br />
=== Description ===<br />
Basic concepts of Lie Groups and algebras including root systems, algebraic groups, and representation theory.<br />
Other topics may be presented.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
[[Math 673]] or equivalent.<br />
<br />
=== Minimal learning outcomes ===<br />
Students will master the basic concepts of Lie Groups and algebras including root systems, algebraic groups, and representation theory.<br />
Possible applications to physics or to other areas of mathematics.<br />
<br />
=== Textbooks ===<br />
<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
Humphreys, James E. Introduction to Lie algebras and representation theory. Second printing, revised. Graduate Texts in Mathematics, 9. Springer-Verlag, New York-Berlin, 1978.<br />
<br />
Bump, Daniel Lie groups. Second edition. Graduate Texts in Mathematics, 225. Springer, New York, 2013. xiv+551<br />
<br />
=== Additional topics ===<br />
Any advanced topic in Algebra including Galois theory, Grobner bases, Representation theory, Invariant theory, etc.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|674]]<br />
None.</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_513R:_Advanced_Topics_in_Applied_Math&diff=2358Math 513R: Advanced Topics in Applied Math2014-12-01T21:11:02Z<p>Roundy: /* Minimal learning outcomes */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Advanced Topics in Applied Mathematics.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Prerequisite ===<br />
instructor's consent<br />
<br />
=== Description ===<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
</div><br />
<br />
Students should gain a familiarity with a particular area of applied mathematics selected by the instructor. In general, a significant, coherent set of readings will be defined, and the student's base of mathematical knowledge and expertise will increase in notable, measurable ways. The students is expected to demonstrate mastery of the material in a manner that is acceptable to the professor.<br />
<br />
=== Textbooks ===<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|513]]</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_438:_Modeling_with_Dynamics_and_Control_2&diff=1931Math 438: Modeling with Dynamics and Control 22012-06-13T19:39:21Z<p>Roundy: /* Title */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Differential and Integral Equations 2<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:1)<br />
<br />
=== Offered ===<br />
W<br />
<br />
=== Prerequisite ===<br />
[[Math 436]], [[Math 402]]; concurrent with [[Math 439]], [[Math 404]]<br />
<br />
=== Description ===<br />
An introduction to the theory of integral equations, of the calculus of variations, of stochastic differential equations and of optimal stochastic control. An introduction to the algorithms that are commonly used to study these systems<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
[[Math 436]], [[Math 402]]; concurrent with [[Math 439]], [[Math 404]]<br />
<br />
=== Minimal learning outcomes ===<br />
Students will have a solid understanding of the concepts listed below. They will be able to prove theorems that are central to this material, including theorems that they have not seen before. They will understand the model specifications for the algorithms, and be able to recognize whether they apply to a given application or not. They will be able to perform the relevant computations on small, simple problems. They will have a basic knowledge of the capabilities of commercial software that available for these problems.<br />
<br />
# Integral Equations<br />
#* Classification and Origins<br />
#* Relationship to Differential Equations<br />
#* Fredholm Equations<br />
#* Symmetric Kernels<br />
#* Volterra Equations<br />
#* General Integral Equations<br />
# Calculus of Variations<br />
#* Variational Problems<br />
#* Euler-Lagrange Condition<br />
#* Second Variation<br />
#* Generalizations of the Variational Problem<br />
#* Hamiltonian Theory<br />
# Optimal Control<br />
#* Problem Formulation<br />
#* Hamilton-Jacobi-Bellman Equation<br />
#* The Adjoint Equation<br />
#* Sufficient Conditions<br />
#* Linear Quadratic Regulator (LQR)<br />
# Stochastic Differential Equations<br />
#* Brownian Motion and Diffusion<br />
#* Weiner Processes<br />
#* Itô Processes and Itô's Lemma<br />
#* Black-Scholes Equation<br />
# Stochastic Optimal Control<br />
#* Problem Formulation<br />
#* Hamilton-Jacobi-Bellman Equation<br />
#* Optimal Stopping Times<br />
#* Linear Quadratic Gaussian (LQG)<br />
#* Investment-Consumption Problems<br />
<br />
<br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|438]]</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_402:_Modeling_with_Uncertainty_and_Data_1&diff=1930Math 402: Modeling with Uncertainty and Data 12012-06-13T19:28:52Z<p>Roundy: /* Offered */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Probability and Statistics 1<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:1)<br />
<br />
=== Offered ===<br />
The course runs through both Spring and Summer<br />
<br />
=== Prerequisite ===<br />
[[Math 322]], [[Math 346]]; concurrent with [[Math 403]]<br />
<br />
=== Description ===<br />
The theory of probability and stochastic processes, emphasizing topics that are used in applications. Random spaces and variables, probability distributions, limit theorems, martingales, diffusion, Markov, Poisson and queuing processes, renewal theory and information theory.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
[[Math 322]], [[Math 346]]; concurrent with [[Math 403]]<br />
<br />
=== Minimal learning outcomes ===<br />
Students will have a solid understanding of the concepts listed below. They will be able to prove theorems that are central to this material, including theorems that they have not seen before. They will understand connections between the concepts taught, and will be able to perform the related computations on small, simple problems. They will understand the model specifications for martingales and for diffusion, Markov, Poisson, queuing and renewal theoretic processes, and be able to recognize whether they apply in the context of a given application or not. They will be able to perform the relevant computations on small, simple problems.<br />
<br />
# Random Spaces and Variables<br />
#* Probability Spaces (including σ-algebras) <br />
#* Random Variables (including Measurable Functions) <br />
#* Expectation (including Lebesgue Integration) <br />
#* Independence<br />
#* Conditional Expectation<br />
#* Law of Large Numbers<br />
# Distributions<br />
#* Generating Functions and Characteristic Functions<br />
#* Moments<br />
#* Commonly Used Distributions<br />
#* Joint and Conditional Distributions<br />
# Limit Theorems<br />
#* Weak Convergence<br />
#* Central Limit Theorem<br />
#* Applications<br />
# Martingales and Diffusion<br />
#* Stochastic Processes, Filtrations, Stopping Times<br />
#* Martingales<br />
#* Doob's Decomposition Theorem<br />
#* Doob's Inequality and Convergence Theorems<br />
# Markov Processes<br />
#* The Markov Property<br />
#* Finite Markov Chains<br />
#* Asymptotic Behavior<br />
#* Absorbing Markov Chains<br />
#* Continuous-Time Markov Chains<br />
# Poisson, Queuing, and Renewal Theory<br />
#* Counting Integrals<br />
#* Kolomogorov's Forward System<br />
#* Poisson Processes<br />
#* Queues<br />
#* Renewal Processes<br />
# Information Theory<br />
#* Entropy<br />
#* Conditional and Joint Entropy<br />
#* Kullback-Lieber Distance<br />
#* Channel Capacity<br />
<br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|402]]</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_402:_Modeling_with_Uncertainty_and_Data_1&diff=1929Math 402: Modeling with Uncertainty and Data 12012-06-13T19:25:55Z<p>Roundy: /* (Credit Hours:Lecture Hours:Lab Hours) */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Probability and Statistics 1<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:1)<br />
<br />
=== Offered ===<br />
The course runs through both the Spring and Summer semesters<br />
<br />
=== Prerequisite ===<br />
[[Math 322]], [[Math 346]]; concurrent with [[Math 403]]<br />
<br />
=== Description ===<br />
The theory of probability and stochastic processes, emphasizing topics that are used in applications. Random spaces and variables, probability distributions, limit theorems, martingales, diffusion, Markov, Poisson and queuing processes, renewal theory and information theory.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
[[Math 322]], [[Math 346]]; concurrent with [[Math 403]]<br />
<br />
=== Minimal learning outcomes ===<br />
Students will have a solid understanding of the concepts listed below. They will be able to prove theorems that are central to this material, including theorems that they have not seen before. They will understand connections between the concepts taught, and will be able to perform the related computations on small, simple problems. They will understand the model specifications for martingales and for diffusion, Markov, Poisson, queuing and renewal theoretic processes, and be able to recognize whether they apply in the context of a given application or not. They will be able to perform the relevant computations on small, simple problems.<br />
<br />
# Random Spaces and Variables<br />
#* Probability Spaces (including σ-algebras) <br />
#* Random Variables (including Measurable Functions) <br />
#* Expectation (including Lebesgue Integration) <br />
#* Independence<br />
#* Conditional Expectation<br />
#* Law of Large Numbers<br />
# Distributions<br />
#* Generating Functions and Characteristic Functions<br />
#* Moments<br />
#* Commonly Used Distributions<br />
#* Joint and Conditional Distributions<br />
# Limit Theorems<br />
#* Weak Convergence<br />
#* Central Limit Theorem<br />
#* Applications<br />
# Martingales and Diffusion<br />
#* Stochastic Processes, Filtrations, Stopping Times<br />
#* Martingales<br />
#* Doob's Decomposition Theorem<br />
#* Doob's Inequality and Convergence Theorems<br />
# Markov Processes<br />
#* The Markov Property<br />
#* Finite Markov Chains<br />
#* Asymptotic Behavior<br />
#* Absorbing Markov Chains<br />
#* Continuous-Time Markov Chains<br />
# Poisson, Queuing, and Renewal Theory<br />
#* Counting Integrals<br />
#* Kolomogorov's Forward System<br />
#* Poisson Processes<br />
#* Queues<br />
#* Renewal Processes<br />
# Information Theory<br />
#* Entropy<br />
#* Conditional and Joint Entropy<br />
#* Kullback-Lieber Distance<br />
#* Channel Capacity<br />
<br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|402]]</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_402:_Modeling_with_Uncertainty_and_Data_1&diff=1928Math 402: Modeling with Uncertainty and Data 12012-06-13T19:25:13Z<p>Roundy: /* Offered */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Probability and Statistics 1<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:1)<br />
<br />
=== Offered<br />
The course runs through both the Spring and Summer semesters<br />
<br />
=== Prerequisite ===<br />
[[Math 322]], [[Math 346]]; concurrent with [[Math 403]]<br />
<br />
=== Description ===<br />
The theory of probability and stochastic processes, emphasizing topics that are used in applications. Random spaces and variables, probability distributions, limit theorems, martingales, diffusion, Markov, Poisson and queuing processes, renewal theory and information theory.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
[[Math 322]], [[Math 346]]; concurrent with [[Math 403]]<br />
<br />
=== Minimal learning outcomes ===<br />
Students will have a solid understanding of the concepts listed below. They will be able to prove theorems that are central to this material, including theorems that they have not seen before. They will understand connections between the concepts taught, and will be able to perform the related computations on small, simple problems. They will understand the model specifications for martingales and for diffusion, Markov, Poisson, queuing and renewal theoretic processes, and be able to recognize whether they apply in the context of a given application or not. They will be able to perform the relevant computations on small, simple problems.<br />
<br />
# Random Spaces and Variables<br />
#* Probability Spaces (including σ-algebras) <br />
#* Random Variables (including Measurable Functions) <br />
#* Expectation (including Lebesgue Integration) <br />
#* Independence<br />
#* Conditional Expectation<br />
#* Law of Large Numbers<br />
# Distributions<br />
#* Generating Functions and Characteristic Functions<br />
#* Moments<br />
#* Commonly Used Distributions<br />
#* Joint and Conditional Distributions<br />
# Limit Theorems<br />
#* Weak Convergence<br />
#* Central Limit Theorem<br />
#* Applications<br />
# Martingales and Diffusion<br />
#* Stochastic Processes, Filtrations, Stopping Times<br />
#* Martingales<br />
#* Doob's Decomposition Theorem<br />
#* Doob's Inequality and Convergence Theorems<br />
# Markov Processes<br />
#* The Markov Property<br />
#* Finite Markov Chains<br />
#* Asymptotic Behavior<br />
#* Absorbing Markov Chains<br />
#* Continuous-Time Markov Chains<br />
# Poisson, Queuing, and Renewal Theory<br />
#* Counting Integrals<br />
#* Kolomogorov's Forward System<br />
#* Poisson Processes<br />
#* Queues<br />
#* Renewal Processes<br />
# Information Theory<br />
#* Entropy<br />
#* Conditional and Joint Entropy<br />
#* Kullback-Lieber Distance<br />
#* Channel Capacity<br />
<br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|402]]</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_655:_Differential_Topology&diff=1858Math 655: Differential Topology2012-06-05T23:04:48Z<p>Roundy: /* Prerequisite */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Differential Topology<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Prerequisite ===<br />
[[Math 342|342]] or equivalent, and [[Math 554|554]] or equivalent.<br />
<br />
=== Description ===<br />
An introduction to manifolds and smooth manifolds and their topology.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
<br />
Knowledge of basic point set topology from [[Math 553]], [[Math 554|554]] will be assumed. This includes topological spaces, basis and countability, metric spaces, quotient spaces, fundamental group, and covering maps. Basic knowledge of linear algebra ([[Math 313]]) and introductory analysis ([[Math 341]] and [[Math 342|342]]) will also be assumed.<br />
<br />
=== Minimal learning outcomes ===<br />
Outlined below are topics that all successful Math 655 students should understand well. Students should be able to demonstrate mastery of relevant vocabulary, and use the vocabulary fluently in their work. They should know common examples and counterexamples, and be able prove that these examples and counterexamples have properties as claimed. Additionally, students should know the content (and limitations) of major theorems and the ideas of the proofs, and apply results of these theorems to solve suitable problems, or use techniques of the proofs to prove additional related results, or to make calculations and computations.<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
# Manifolds<br />
#* Topological and smooth manifolds<br />
#* Manifolds with boundary<br />
#* Tangent vectors<br />
#* Tangent bundles<br />
#* Vector bundles and bundle maps<br />
#* Cotangent bundles<br />
# Submanifolds<br />
#* Submersions, immersions, embeddings<br />
#* Inverse and implicit function theorems<br />
#* Transversality<br />
#* Embedding and approximation theorems<br />
# Differential forms and tensors<br />
#* Wedge product<br />
#* Exterior derivative<br />
#* Orientations<br />
#* Stoke's Theorem<br />
<br />
</div><br />
<br />
=== Additional topics ===<br />
<br />
At the discretion of the instructor as time allows. Topics might include Lie groups and homogeneous spaces, Morse functions, de Rham cohomology and the de Rham theorem, Jordan curve theorem, Lefschetz fixed-point theory, degree, Gauss-Bonnet theorem, etc.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
[[Math 656]]<br />
<br />
[[Category:Courses|655]]</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_314:_Calculus_of_Several_Variables&diff=1857Math 314: Calculus of Several Variables2012-06-05T22:56:21Z<p>Roundy: /* Prerequisite */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Calculus of Several Variables.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
F, W, Sp, Su<br />
<br />
=== Prerequisite ===<br />
[[Math 113]]; [[Math 313|313]].<br />
<br />
=== Description ===<br />
Partial differentiation, the Jacobian matrix, and integral theorems of vector calculus.<br />
<br />
== Desired Learning Outcomes ==<br />
This course is aimed at students majoring in mathematical and physical sciences, and engineering, and students minoring in mathematics or mathematical education. Calculus is the foundation for most of the mathematics studied at the university level. The mastery of calculus requires well-developed manipulative skills, clear conceptual understanding, and the ability to model phenomena in a variety of settings. Calculus of several variables extends the concepts of limit, integral, and derivative from one dimension to higher dimensional settings and is therefore fundamental for many fields of mathematics. This course contributes to all the expected learning outcomes of the Mathematics BS (see [[http://learningoutcomes.byu.edu]]).<br />
<br />
=== Prerequisites ===<br />
Students are expected to have completed [[Math 113]], and [[Math 313]] or concurrent registration in [[Math 313]].<br />
<br />
=== Minimal learning outcomes ===<br />
Students should achieve mastery of the topics below. This means that they should know all relevant definitions, full statements of the major theorems, and examples of the various concepts. Further, students should be able to solve non-trivial problems related to these concepts, and prove simple theorems in analogy to proofs given by the instructor.<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
# Vectors and vector functions<br />
#* Compute vector operations<br />
#* Describe lines and planes in space<br />
#* Identify standard quadratic surfaces<br />
#* Describe parametric curves as vector functions<br />
#* Compute and interpret the derivative of a vector function<br />
#* Apply vector functions to curvilinear motion<br />
# Derivatives of functions of several variables<br />
#* Compute and interpret partial derivatives<br />
#* Use the gradient to find directional derivatives and normal vectors<br />
#* Find local and global extrema<br />
#* Solve optimization problems involving several variables<br />
#* Use the method of Lagrange to find extrema under constraints<br />
# Multiple integrals<br />
#* Know and apply Fubini’s theorem to express multiple integrals as iterated integrals<br />
#* Change the order of integration<br />
#* Transform integrals to polar, cylindrical, and spherical coordinates<br />
#* Use multiple integrals to find area, volume, mass, center of mass, and other applications<br />
#* Use the Jacobian of a coordinate change to transform integrals<br />
# Line and surface integrals<br />
#* Evaluate line integrals<br />
#* Use line integrals to compute work and circulation<br />
#* Use surface integrals to compute surface area and flux<br />
#* Set up and use integrals over parametric surfaces<br />
# Divergence and curl<br />
#* Explain and interpret divergence and curl<br />
#* Know and apply the divergence theorem<br />
#* Know and apply Stokes’ theorem<br />
#* Use the divergence and curl tests to describe properties of vector fields<br />
<br />
</div><br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
These are at the instructor's discretion as time allows; applications to physical problems are particularly helpful.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
This course is required for [[Math 303]], [[Math 352]], [[Math 410]], [[Math 465]], [[Math 480]], [[Math 541]], [[Math 543]], and [[Math 547]].<br />
<br />
[[Category:Courses|314]]</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_522:_Methods_of_Applied_Math_2&diff=1838Math 522: Methods of Applied Math 22012-05-10T20:20:48Z<p>Roundy: </p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Methods of Applied Mathematics 2.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Prerequisite ===<br />
[[Math 334]] or equivalents.<br />
<br />
=== Description ===<br />
Possible topics include variational, integral, and partial differential equations; spectral and transform methods; nonlinear waves; Green's functions; scaling and asymptotic analysis; perturbation theory; continuum<br />
mechanics.<br />
<br />
== Desired Learning Outcomes ==<br />
The object of this course is to familiarize students with classical techniques in applied mathematics and demonstrate their application to specific problems. <br />
=== Prerequisites ===<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
</div><br />
=== Textbooks ===<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|522]]</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_521:_Methods_of_Applied_Math_1&diff=1837Math 521: Methods of Applied Math 12012-05-10T20:20:28Z<p>Roundy: </p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Methods of Applied Mathematics 1.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Prerequisite ===<br />
[[Math 334]] or equivalents.<br />
<br />
=== Description ===<br />
Possible topics include variational, integral, and partial differential equations; spectral and transform methods; nonlinear waves; Green's functions; scaling and asymptotic analysis; perturbation theory; continuum<br />
mechanics.<br />
<br />
== Desired Learning Outcomes ==<br />
The object of this course is to familiarize students with classical techniques in applied mathematics and demonstrate their application to specific problems. The list above gives examples of possible techniques and problems. <br />
=== Prerequisites ===<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
</div><br />
=== Textbooks ===<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|521]]</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_511:_Numerical_Methods_for_PDEs&diff=1836Math 511: Numerical Methods for PDEs2012-05-10T20:20:08Z<p>Roundy: </p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Numerical Methods for Partial Differential Equations.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Prerequisite ===<br />
[[Math 303]] or [[Math 347|347]]; [[Math 410|410]]; or equivalents.<br />
<br />
=== Description ===<br />
Finite difference and finite volume methods for partial differential equations. Stability, consistency, and convergence theory.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
This course is designed to prepare students to solve mathematical<br />
models represented by initial or boundary value problems involving<br />
partial differential equations that cannot be solved directly<br />
using standard mathematical techniques but are amenable to a<br />
computational approach. Numerical solution of partial differential equations has important applications in many application areas. Students are introduced to the<br />
discretization methodologies, with particular emphasis on the<br />
finite difference method, that allows the construction of<br />
accurate and stable numerical schemes. In depth discussion of<br />
theoretical aspects such as stability analysis and convergence<br />
will be used to enhance the students' understanding of the<br />
numerical methods. Students will also be required to perform some<br />
programming and computation so as to gain experience in<br />
implementing the schemes and to be able to observe the numerical<br />
performance of the various numerical methods.<br />
<br />
The course addresses the University goal of developing the skills<br />
of sound thinking, effective communication and quantitative<br />
reasoning. The course also allow students, especially<br />
undergraduate students, to develop some depth and consequently<br />
competence in an important area of applied mathematics.<br />
<br />
This course requires knowledge of higher level courses in<br />
mathematics and serves as an introductory graduate<br />
level course to prepare the students to apply the methods learned<br />
in their research projects.<br />
<br />
=== Prerequisites ===<br />
<br />
Understanding of basic theory and properties of solutions of partial differential equations;<br />
<br />
Basic programming skill in matlab;<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
</div><br />
Students are expected to acquire the following knowledge and skills:<br />
<br />
Derive finite difference schemes using Taylor series.<br />
<br />
Derive finite volume schemes using flux balance.<br />
<br />
Understand how finite volume scheme and finite difference scheme are related.<br />
<br />
Determine the consistency of a difference scheme.<br />
<br />
Explain the proper function spaces and discrete norms for<br />
grid functions for use in analysis of stability.<br />
<br />
Establish the stability of a difference scheme using <br />
(1) Heuristic approach <br />
(2) Energy method <br />
(3) von Neumann method <br />
(4) Matrix method.<br />
<br />
Recall the CFL condition its relation with stability.<br />
<br />
Explain the convergence of the finite difference<br />
approximations and its relation with consistency and stability via<br />
Lax theorem;<br />
<br />
Determine the order of accuracy of a finite difference<br />
scheme.<br />
<br />
Implement finite difference schemes on computers and perform<br />
numerical studies of the stability and convergence properties of<br />
the schemes.<br />
<br />
Explain the role and the control of numerical diffusion and<br />
dispersion in computation ; to determine how numerical phase speed<br />
and group velocity may deviate from the theoretical phase speed<br />
and group velocity and the numerical techniques to handle such<br />
issues.<br />
<br />
Recall numerical methods that efficiently handle a<br />
multidimensional problem.<br />
<br />
Recall alternating direction methods that reduce higher<br />
dimensional problems into a sequence of one dimensional problems.<br />
<br />
Recall the maximum principles for numerical schemes for<br />
Laplace equations.<br />
<br />
Recall iterative techniques for solving the linear systems<br />
resulting from finite difference or finite element discretization.<br />
<br />
=== Textbooks ===<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
John Strikwerda, Finite Difference Schemes and Partial Differential Equations, 2nd Ed., SIAM, 2007;<br />
ISBN: 089871639X, 978-0898716399<br />
<br />
Randall Leveque, Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems, SIAM 2007;<br />
ISBN: 0898716292, 978-0898716290<br />
<br />
K. W. Morton and D. F. Mayers, Numerical Solution of Partial Differential Equations: An Introduction, 2nd Ed., Cambridge University Press, 2005;<br />
ISBN: 0521607930, 978-0521607933<br />
<br />
Arieh Iserles, A First Course in the Numerical Analysis of Differential Equations, 2nd Ed, Cambridge University Press, 2008;<br />
ISBN: 0521734908, 978-0521734905<br />
<br />
Claes Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Dover, 2009;<br />
ISBN: 048646900X, 978-0486469003<br />
<br />
J.W. Thomas, Numerical Partial Differential Equations: Finite Difference Methods, 2nd Ed., Springer, 2010;<br />
ISBN-10: 1441931058, 978-1441931054<br />
<br />
=== Additional topics ===<br />
Finite element method; Method of lines; Parallel computing<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|511]]<br />
Math 303 or 347; 410; or equivalents.</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_510:_Numerical_Methods_for_Linear_Algebra&diff=1835Math 510: Numerical Methods for Linear Algebra2012-05-10T20:19:51Z<p>Roundy: </p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Numerical Methods for Linear Algebra.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Prerequisite ===<br />
[[Math 343]], [[Math 410|410]]; or equivalents.<br />
<br />
=== Description ===<br />
Numerical matrix algebra, orthogonalization and least squares methods, unsymmetric and symmetric eigenvalue problems, iterative<br />
methods, advanced solvers for partial differential equations.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
This course is designed to prepare students to solve linear algebra<br />
problems arising from many applications such as mathematical models of physical or engineering processes.<br />
Students are introduced to modern concepts and methodologies in numerical linear algebra, with particular emphasis on t<br />
methods that can be used to solve very large scale problems. In depth discussion of<br />
theoretical aspects such as stability and convergence<br />
will be used to enhance the students' understanding of the<br />
numerical methods. Students will also be required to perform some<br />
programming and computation so as to gain experience in<br />
implementing and observing the numerical<br />
performance of the various numerical methods.<br />
<br />
The course addresses the University goal of developing the skills<br />
of sound thinking, effective communication and quantitative<br />
reasoning. The course also allow students, especially<br />
undergraduate students, to develop some depth and consequently<br />
competence in an important area of applied mathematics.<br />
<br />
This course requires knowledge of higher level courses in<br />
mathematics. The course also serves as an introductory graduate<br />
level course to prepare the students to apply the methods learned<br />
in their research projects.<br />
<br />
=== Prerequisites ===<br />
Mastery of materials in an undergraduate course in linear algebra. Knowledge of matlab.<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
</div><br />
(Must know)<br />
<br />
Know properties of unitary matrices<br />
<br />
Know general and practical definitions and properties of norms<br />
<br />
Know definition and properties of SVD <br />
<br />
Know definitions of and properties of projectors and orthogonal projectors<br />
<br />
Be able to state and apply classical Gram-Schmidt and modified Gram-Schmidt algorithms<br />
<br />
Know definition and properties of a Householder reflector<br />
<br />
Know how to construct Householder QR factorization <br />
<br />
Know how least squares problems arise from a polynomial fitting problem<br />
<br />
Know how to solve least square problems using <br />
(1) normal equations/pseudoinverse, <br />
(2) QR factorization and <br />
(3) SVD<br />
<br />
Be able to define the condition of a problem and related condition number<br />
Know how to calculate the condition number of a matrix <br />
<br />
Understand the concepts of well-conditioned and ill-conditioned problems<br />
<br />
Know how to derive the conditioning bounds<br />
<br />
Know the precise definition of stability and backward stability<br />
<br />
Be able to apply the fundamental axiom of floating point arithmetic to determine stability <br />
<br />
Know the difference between stability and conditioning<br />
<br />
Know the four condition numbers of a least squares problem <br />
<br />
Know how to construct LU and PLU factorizations<br />
<br />
Know how PLU is related to Gaussian elimination<br />
<br />
Understand Cholesky decomposition <br />
<br />
Know properties of eigenvalues and eigenvectors under similarity transformation and shift<br />
<br />
Know various matrix decomposition related to eigenvalue calculation: <br />
(1) spectral decomposition<br />
(2) unitary diagonaliation<br />
(2) Schur decomposition<br />
Understand why and how matrices can be reduced to Hessenberg form <br />
<br />
Know various form of power method and what Rayleigh quotient iteration and their properties and convergence rates<br />
<br />
Know the QR algorithm with shifts <br />
<br />
Understand simultaneous iteration and QR algorithm are mathematically equivalent <br />
<br />
Understand the Arnoldi algorithm and its properties<br />
<br />
Know the polynomial approximation problem associated with Arnoldi method<br />
<br />
Be able to state the GMRES algorithm<br />
<br />
Be able to state three term recurrence of Lanczos iteration for real symmetric matrices<br />
<br />
Understand the CG algorithm and its properties<br />
<br />
Know the v-cycle multigrid algorithm and the full multigrid algorithm<br />
<br />
Know how to construct preconditioners: (block) diagonal, incomplete LU and incomplete Cholesky preconditioners<br />
<br />
Know CGN and BCG and other Krylov space methods<br />
<br />
=== Textbooks ===<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
Lloyd N. Trefethen and David Bau III, Numerical Linear Algebra, SIAM; 1997; <br />
ISBN: 0898713617, 978-0898713619<br />
<br />
James W. Demmel, Applied Numerical Linear Algebra, SIAM; 1997; <br />
ISBN: 0898713897, 978-0898713893<br />
<br />
Gene H. Golub and Charles F. Van Loan, Matrix Computations, 3rd Ed., Johns Hopkins University Press, 1996;<br />
ISBN: 0801854148, 978-0801854149<br />
<br />
William L. Briggs, Van Emden Henson and Steve F. McCormick, A Multigrid Tutorial, 2nd Ed., SIAM, 2000;<br />
ISBN: 0898714621, 978-0898714623<br />
<br />
Barry Smith, Petter Bjorstad, William Gropp, Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations, Cambridge University Press, 2004;<br />
ISBN: 0521602866, 978-0521602860<br />
<br />
=== Additional topics ===<br />
Multigrid method;<br />
Domain decomposition method;<br />
Freely available linear algebra software;<br />
Fast multipole method for linear systems; Parallel processing<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|510]]<br />
Math 343, 410; or equivalents.</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_522:_Methods_of_Applied_Math_2&diff=1834Math 522: Methods of Applied Math 22012-05-10T20:17:17Z<p>Roundy: </p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Methods of Applied Mathematics 2.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Offered ===<br />
On Demand<br />
<br />
=== Prerequisite ===<br />
[[Math 334]] or equivalents.<br />
<br />
=== Description ===<br />
Possible topics include variational, integral, and partial differential equations; spectral and transform methods; nonlinear waves; Green's functions; scaling and asymptotic analysis; perturbation theory; continuum<br />
mechanics.<br />
<br />
== Desired Learning Outcomes ==<br />
The object of this course is to familiarize students with classical techniques in applied mathematics and demonstrate their application to specific problems. <br />
=== Prerequisites ===<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
</div><br />
=== Textbooks ===<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|522]]</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_521:_Methods_of_Applied_Math_1&diff=1833Math 521: Methods of Applied Math 12012-05-10T20:16:29Z<p>Roundy: </p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Methods of Applied Mathematics 1.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Offered ===<br />
On Demand<br />
<br />
=== Prerequisite ===<br />
[[Math 334]] or equivalents.<br />
<br />
=== Description ===<br />
Possible topics include variational, integral, and partial differential equations; spectral and transform methods; nonlinear waves; Green's functions; scaling and asymptotic analysis; perturbation theory; continuum<br />
mechanics.<br />
<br />
== Desired Learning Outcomes ==<br />
The object of this course is to familiarize students with classical techniques in applied mathematics and demonstrate their application to specific problems. The list above gives examples of possible techniques and problems. <br />
=== Prerequisites ===<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
</div><br />
=== Textbooks ===<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|521]]</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_511:_Numerical_Methods_for_PDEs&diff=1832Math 511: Numerical Methods for PDEs2012-05-10T20:14:47Z<p>Roundy: </p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Numerical Methods for Partial Differential Equations.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Offered ===<br />
Winter<br />
<br />
=== Prerequisite ===<br />
[[Math 303]] or [[Math 347|347]]; [[Math 410|410]]; or equivalents.<br />
<br />
=== Description ===<br />
Finite difference and finite volume methods for partial differential equations. Stability, consistency, and convergence theory.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
This course is designed to prepare students to solve mathematical<br />
models represented by initial or boundary value problems involving<br />
partial differential equations that cannot be solved directly<br />
using standard mathematical techniques but are amenable to a<br />
computational approach. Numerical solution of partial differential equations has important applications in many application areas. Students are introduced to the<br />
discretization methodologies, with particular emphasis on the<br />
finite difference method, that allows the construction of<br />
accurate and stable numerical schemes. In depth discussion of<br />
theoretical aspects such as stability analysis and convergence<br />
will be used to enhance the students' understanding of the<br />
numerical methods. Students will also be required to perform some<br />
programming and computation so as to gain experience in<br />
implementing the schemes and to be able to observe the numerical<br />
performance of the various numerical methods.<br />
<br />
The course addresses the University goal of developing the skills<br />
of sound thinking, effective communication and quantitative<br />
reasoning. The course also allow students, especially<br />
undergraduate students, to develop some depth and consequently<br />
competence in an important area of applied mathematics.<br />
<br />
This course requires knowledge of higher level courses in<br />
mathematics and serves as an introductory graduate<br />
level course to prepare the students to apply the methods learned<br />
in their research projects.<br />
<br />
=== Prerequisites ===<br />
<br />
Understanding of basic theory and properties of solutions of partial differential equations;<br />
<br />
Basic programming skill in matlab;<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
</div><br />
Students are expected to acquire the following knowledge and skills:<br />
<br />
Derive finite difference schemes using Taylor series.<br />
<br />
Derive finite volume schemes using flux balance.<br />
<br />
Understand how finite volume scheme and finite difference scheme are related.<br />
<br />
Determine the consistency of a difference scheme.<br />
<br />
Explain the proper function spaces and discrete norms for<br />
grid functions for use in analysis of stability.<br />
<br />
Establish the stability of a difference scheme using <br />
(1) Heuristic approach <br />
(2) Energy method <br />
(3) von Neumann method <br />
(4) Matrix method.<br />
<br />
Recall the CFL condition its relation with stability.<br />
<br />
Explain the convergence of the finite difference<br />
approximations and its relation with consistency and stability via<br />
Lax theorem;<br />
<br />
Determine the order of accuracy of a finite difference<br />
scheme.<br />
<br />
Implement finite difference schemes on computers and perform<br />
numerical studies of the stability and convergence properties of<br />
the schemes.<br />
<br />
Explain the role and the control of numerical diffusion and<br />
dispersion in computation ; to determine how numerical phase speed<br />
and group velocity may deviate from the theoretical phase speed<br />
and group velocity and the numerical techniques to handle such<br />
issues.<br />
<br />
Recall numerical methods that efficiently handle a<br />
multidimensional problem.<br />
<br />
Recall alternating direction methods that reduce higher<br />
dimensional problems into a sequence of one dimensional problems.<br />
<br />
Recall the maximum principles for numerical schemes for<br />
Laplace equations.<br />
<br />
Recall iterative techniques for solving the linear systems<br />
resulting from finite difference or finite element discretization.<br />
<br />
=== Textbooks ===<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
John Strikwerda, Finite Difference Schemes and Partial Differential Equations, 2nd Ed., SIAM, 2007;<br />
ISBN: 089871639X, 978-0898716399<br />
<br />
Randall Leveque, Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems, SIAM 2007;<br />
ISBN: 0898716292, 978-0898716290<br />
<br />
K. W. Morton and D. F. Mayers, Numerical Solution of Partial Differential Equations: An Introduction, 2nd Ed., Cambridge University Press, 2005;<br />
ISBN: 0521607930, 978-0521607933<br />
<br />
Arieh Iserles, A First Course in the Numerical Analysis of Differential Equations, 2nd Ed, Cambridge University Press, 2008;<br />
ISBN: 0521734908, 978-0521734905<br />
<br />
Claes Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Dover, 2009;<br />
ISBN: 048646900X, 978-0486469003<br />
<br />
J.W. Thomas, Numerical Partial Differential Equations: Finite Difference Methods, 2nd Ed., Springer, 2010;<br />
ISBN-10: 1441931058, 978-1441931054<br />
<br />
=== Additional topics ===<br />
Finite element method; Method of lines; Parallel computing<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|511]]<br />
Math 303 or 347; 410; or equivalents.</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_510:_Numerical_Methods_for_Linear_Algebra&diff=1831Math 510: Numerical Methods for Linear Algebra2012-05-10T20:14:02Z<p>Roundy: </p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Numerical Methods for Linear Algebra.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Offered ===<br />
Fall<br />
<br />
=== Prerequisite ===<br />
[[Math 343]], [[Math 410|410]]; or equivalents.<br />
<br />
=== Description ===<br />
Numerical matrix algebra, orthogonalization and least squares methods, unsymmetric and symmetric eigenvalue problems, iterative<br />
methods, advanced solvers for partial differential equations.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
This course is designed to prepare students to solve linear algebra<br />
problems arising from many applications such as mathematical models of physical or engineering processes.<br />
Students are introduced to modern concepts and methodologies in numerical linear algebra, with particular emphasis on t<br />
methods that can be used to solve very large scale problems. In depth discussion of<br />
theoretical aspects such as stability and convergence<br />
will be used to enhance the students' understanding of the<br />
numerical methods. Students will also be required to perform some<br />
programming and computation so as to gain experience in<br />
implementing and observing the numerical<br />
performance of the various numerical methods.<br />
<br />
The course addresses the University goal of developing the skills<br />
of sound thinking, effective communication and quantitative<br />
reasoning. The course also allow students, especially<br />
undergraduate students, to develop some depth and consequently<br />
competence in an important area of applied mathematics.<br />
<br />
This course requires knowledge of higher level courses in<br />
mathematics. The course also serves as an introductory graduate<br />
level course to prepare the students to apply the methods learned<br />
in their research projects.<br />
<br />
=== Prerequisites ===<br />
Mastery of materials in an undergraduate course in linear algebra. Knowledge of matlab.<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
</div><br />
(Must know)<br />
<br />
Know properties of unitary matrices<br />
<br />
Know general and practical definitions and properties of norms<br />
<br />
Know definition and properties of SVD <br />
<br />
Know definitions of and properties of projectors and orthogonal projectors<br />
<br />
Be able to state and apply classical Gram-Schmidt and modified Gram-Schmidt algorithms<br />
<br />
Know definition and properties of a Householder reflector<br />
<br />
Know how to construct Householder QR factorization <br />
<br />
Know how least squares problems arise from a polynomial fitting problem<br />
<br />
Know how to solve least square problems using <br />
(1) normal equations/pseudoinverse, <br />
(2) QR factorization and <br />
(3) SVD<br />
<br />
Be able to define the condition of a problem and related condition number<br />
Know how to calculate the condition number of a matrix <br />
<br />
Understand the concepts of well-conditioned and ill-conditioned problems<br />
<br />
Know how to derive the conditioning bounds<br />
<br />
Know the precise definition of stability and backward stability<br />
<br />
Be able to apply the fundamental axiom of floating point arithmetic to determine stability <br />
<br />
Know the difference between stability and conditioning<br />
<br />
Know the four condition numbers of a least squares problem <br />
<br />
Know how to construct LU and PLU factorizations<br />
<br />
Know how PLU is related to Gaussian elimination<br />
<br />
Understand Cholesky decomposition <br />
<br />
Know properties of eigenvalues and eigenvectors under similarity transformation and shift<br />
<br />
Know various matrix decomposition related to eigenvalue calculation: <br />
(1) spectral decomposition<br />
(2) unitary diagonaliation<br />
(2) Schur decomposition<br />
Understand why and how matrices can be reduced to Hessenberg form <br />
<br />
Know various form of power method and what Rayleigh quotient iteration and their properties and convergence rates<br />
<br />
Know the QR algorithm with shifts <br />
<br />
Understand simultaneous iteration and QR algorithm are mathematically equivalent <br />
<br />
Understand the Arnoldi algorithm and its properties<br />
<br />
Know the polynomial approximation problem associated with Arnoldi method<br />
<br />
Be able to state the GMRES algorithm<br />
<br />
Be able to state three term recurrence of Lanczos iteration for real symmetric matrices<br />
<br />
Understand the CG algorithm and its properties<br />
<br />
Know the v-cycle multigrid algorithm and the full multigrid algorithm<br />
<br />
Know how to construct preconditioners: (block) diagonal, incomplete LU and incomplete Cholesky preconditioners<br />
<br />
Know CGN and BCG and other Krylov space methods<br />
<br />
=== Textbooks ===<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
Lloyd N. Trefethen and David Bau III, Numerical Linear Algebra, SIAM; 1997; <br />
ISBN: 0898713617, 978-0898713619<br />
<br />
James W. Demmel, Applied Numerical Linear Algebra, SIAM; 1997; <br />
ISBN: 0898713897, 978-0898713893<br />
<br />
Gene H. Golub and Charles F. Van Loan, Matrix Computations, 3rd Ed., Johns Hopkins University Press, 1996;<br />
ISBN: 0801854148, 978-0801854149<br />
<br />
William L. Briggs, Van Emden Henson and Steve F. McCormick, A Multigrid Tutorial, 2nd Ed., SIAM, 2000;<br />
ISBN: 0898714621, 978-0898714623<br />
<br />
Barry Smith, Petter Bjorstad, William Gropp, Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations, Cambridge University Press, 2004;<br />
ISBN: 0521602866, 978-0521602860<br />
<br />
=== Additional topics ===<br />
Multigrid method;<br />
Domain decomposition method;<br />
Freely available linear algebra software;<br />
Fast multipole method for linear systems; Parallel processing<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|510]]<br />
Math 343, 410; or equivalents.</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_313:_Elementary_Linear_Algebra&diff=1830Math 313: Elementary Linear Algebra2012-05-10T20:06:41Z<p>Roundy: /* Prerequisite */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Elementary Linear Algebra.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
F, W, Sp, Su<br />
<br />
=== Prerequisite ===<br />
[[Math 112]]. Students are recommended to take [[Math 290]] before taking Math 313<br />
<br />
=== Description ===<br />
Linear systems, matrices, vectors and vector spaces, linear transformations, determinants, inner product spaces, eigenvalues, and eigenvectors.<br />
<br />
This course is aimed at majors in mathematics, the physical sciences, engineering, and other students interested in applications of mathematics to their disciplines. Linear algebra is used more than any other form of advanced mathematics in industry and science. A key idea is the mathematical modeling of a problem via systems of linear equations.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
<br />
[[Math 113]]. [[Math 290]] is encouraged.<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
Upon completion of this course, the successful student will be able to:<br />
<br />
# Use Gaussian elimination to do all of the following: solve a linear system with reduced row echelon form, solve a linear system with row echelon form and backward substitution, find the inverse of a given matrix, and find the determinant of a given matrix.<br />
# Demonstrate proficiency at matrix algebra. For matrix multiplication demonstrate understanding of the associative law, the reverse order law for inverses and transposes, and the failure of the commutative law and the cancellation law.<br />
# Use Cramer's rule to solve a linear system.<br />
# Use cofactors to find the inverse of a given matrix and the determinant of a given matrix.<br />
# Determine whether a set with a given notion of addition and scalar multiplication is a vector space. Here, and in relevant numbers below, be familiar with both finite and infinite dimensional examples.<br />
# Determine whether a given subset of a vector space is a subspace.<br />
# Determine whether a given set of vectors is linearly independent, spans, or is a basis.<br />
# Determine the dimension of a given vector space or of a given subspace.<br />
# Find bases for the null space, row space, and column space of a given matrix, and determine its rank.<br />
# Demonstrate understanding of the Rank-Nullity Theorem and its applications.<br />
# Given a description of a linear transformation, find its matrix representation relative to given bases.<br />
# Demonstrate understanding of the relationship between similarity and change of basis.<br />
# Determine whether a real valued function on a vector space is a norm.<br />
# Find the norm of a vector and the angle between two vectors in an inner product space.<br />
# Use the inner product to express a vector in an inner product space as a linear combination of an orthogonal set of vectors.<br />
# Find the orthogonal complement of a given subspace.<br />
# Demonstrate understanding of the relationship of the row space, column space, and nullspace of a matrix (and its transpose) via orthogonal complements.<br />
# Demonstrate understanding of the Cauchy-Schwartz inequality and its applications.<br />
# Determine whether a vector space with a (sesquilinear) form is an inner product space.<br />
# Use the Gram-Schmidt process to find an orthonormal basis of an inner product space. Be capable of doing this both in '''R'''<sup>n</sup> and in function spaces that are inner product spaces.<br />
# Use least squares to fit a line (''y'' = ''ax'' + ''b'') to a table of data, plot the line and data points, and explain the meaning of least squares in terms of orthogonal projection.<br />
# Use the idea of least squares to find orthogonal projections onto subspaces and for polynomial curve fitting.<br />
# Find (real and complex) eigenvalues and eigenvectors of 2 &times; 2 or 3 &times; 3 matrices.<br />
# Determine whether a given matrix is diagonalizable. If so, find a matrix that diagonalizes it via similarity.<br />
# Demonstrate understanding of the relationship between eigenvalues of a square matrix and its determinant, its trace, and its invertibility/singularity.<br />
# Identify symmetric matrices and orthogonal matrices.<br />
# Find a matrix that orthogonally diagonalizes a given symmetric matrix.<br />
# Know and be able to apply the spectral theorem for symmetric matrices.<br />
# Correctly define terms and give examples relating to the above concepts.<br />
# Prove basic theorems about the above concepts.<br />
# Prove or disprove statements relating to the above concepts.<br />
# Be adept at hand computation for row reduction, matrix inversion and similar problems; also, use MATLAB or a similar program for linear algebra problems.<br />
<br />
<br />
<br />
</div><br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Math 314]], [[Math 334]], [[Math 342]], [[Math 355]], [[Math 371]], [[Math 431]], [[Math 485]], [[Math 570]]. It is clear from this list that it is imperative to cover all the minimal learning outcomes.<br />
<br />
[[Category:Courses|313]]</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_119&diff=1829Math 1192012-05-10T20:05:22Z<p>Roundy: /* Offered */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Introduction to Calculus.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(4:4:1)<br />
<br />
=== Offered ===<br />
F, W, Sp, Su - BEGINNING FALL 2011 THIS COURSE WILL ONLY BE AVAILABLE THROUGH INDEPENDENT STUDY.<br />
<br />
=== Prerequisite ===<br />
[[Math 110]] or equivalent.<br />
<br />
=== Description ===<br />
Introduction to plane analytic geometry and calculus.<br />
<br />
=== Note ===<br />
Effective Fall 2011: The math department recommends that students considering taking Math 119 should instead consider taking Math 112 (for Life Sciences students) or Math 118 (for students in the Marriott School of Management). Beginning Fall 2011 Math 119 will be available as an evening course through the department of continuing education.<br />
<br />
== Desired Learning Outcomes ==<br />
This course is designed to provide beginning undergraduate students a first exposure to Calculus concepts, theorems, and techniques. In general, theorems will not be proved. An understanding of the theorems and their applications to solve calculus problems will be taught. Students will be expected to develop effective problem solving skills based on their understanding of the theory and not just memorize a set of routines to solve problems.<br />
<br />
=== Prerequisites ===<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
# Determine limits of functions from their graphs or equations.<br />
# Analyze and apply the notions of continuity and differentiability to functions.<br />
# Calculate derivatives for a variety of simple functions and use the concept of derivative to solve problems from various applications.<br />
# Use derivatives to construct and analyze graphs of selected functions.<br />
# Use various techniques to determine antiderivatives of simple functions.<br />
# Demonstrate the connection between area and the definite integral.<br />
# Apply the Fundamental Theorem of Calculus to evaluate definite integrals.<br />
# Integrate selected functions and apply the concepts of integration to solve various applications.</div><br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
A brief introduction to multivariable calculus, solving simple differential equations, topics in probability and statistics, sequences, and series.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
This course is not a prerequisite for any math course at Brigham Young University.<br />
[[Category:Courses|119]]</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_118:_Finite_Mathematics&diff=1828Math 118: Finite Mathematics2012-05-10T20:04:28Z<p>Roundy: /* Offered */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Finite Mathematics<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
Fall, Winter, Spring, Summer<br />
<br />
=== Prerequisite ===<br />
Math 110<br />
<br />
=== Description ===<br />
Description: This course studies the basic elements and applications of finite mathematics. The first half of the class covers the language of set theory, principles of counting and combinatorics, probability theory for equally likely outcomes, elementary stochastic processes, conditional probabilities, and repeated experiments. The concept of a random variable is developed, along with expectation and variance. The second half of the class explores systems of linear equations and matrix algebra, linear programming, and Markov chains. This course considers a broad range of applications in business, the life sciences, and the social sciences.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
# Part 1: Probability Models (20 lectures, approx.)<br />
#* Sets Theory (3 lectures)<br />
#** Describe sets using set-builder notation.<br />
#** Solve problems involving set membership, subsets, intersections, unions, and complements of sets.<br />
#** Be able to identify the various partitions of a Venn diagram.<br />
#** Determine the number of elements in a partition, based on set counting rules for unions, complements, and products.<br />
#* Combinatorics & Counting (6 lectures)<br />
#** Describe the sample space of an experiment and the set of all possible outcomes using trees and the multiplicative principle, as appropriate.<br />
#** Explain what a permutation is, and how many permutations there are for a given set.<br />
#** Demonstrate how to count more sophisticated permutation problems involving products and restrictions to sets.<br />
#** Use the notion of partitions to reduce permutation problems to combinations, where the order does not matter.<br />
#** Be able to solve various hybrid counting problems with and without replacement, with and without order.<br />
#** Be able to apply ideas from combinatorics and counting, and formulate real-world problems.<br />
#* Probability (8 lectures)<br />
#** Be able to describe the notions of outcomes and events in probability, and the axioms of a probability space.<br />
#** Use ideas from combinatorics to determine the probabilities of various events, with equally likely outcomes.<br />
#** Demonstrate understanding of probabilistic independence.<br />
#** Describe a stochastic process, and be able to compute probabilities of events on trees.<br />
#** Explain Bayes rule and demonstrate proficiency with conditional probability.<br />
#** Be able to use Bayes formula to compute probabilities of various conditionals.<br />
#** Be proficient with Bernoulli trials, be able to solve basic problems.<br />
#** Be able to apply ideas from probability, and formulate real-world problems.<br />
#* Random Variables, Expected Values, Variance (3 lectures)<br />
#** Demonstrate understanding of what a random variable is. Describe the probability density function and the distribution of a given random variable.<br />
#** Show how to compute the expectation, variance, and standard deviation of a random variable.<br />
#** Be able to read a table for a normal distribution and compute the probabilities of given events.<br />
#** Be able to apply ideas of uncertainty, and formulate real-world problems.<br />
# Part 2: Linear Models (20 lectures, approx.)<br />
#* Systems of Linear Equations (3 lectures)<br />
#** Use elimination and substitution methods to solve linear systems of two or three variables.<br />
#** Reduce a linear system into row echelon form, then solve with back substitution.<br />
#** Solve a linear system by transforming it into reduced row echelon form.<br />
#** Identify whether a system of linear equations has no solution, exactly one solution, or infinitely many solutions.<br />
#* Matrix Algebra and Applications (3 lectures)<br />
#** Perform matrix algebra operations: addition, multiplication.<br />
#** Compute the inverse of a matrix, identify when a matrix is not invertible.<br />
#** Study in detail at least one application involving matrix algebra (e.g., Leontief economic models).<br />
#* Linear Programming (8 lectures)<br />
#** Formulate linear programming problems from various application areas, such as business, resource management, etc.<br />
#** Describe the constraints, the feasible set, and the objective function of a given linear optimization problem.<br />
#** Solve a linear program using the graphical method, when possible.<br />
#** Explain the standard form of a linear program.<br />
#** Describe the following concepts from the simplex method: slack variable, pivot column, tableau.<br />
#** Be able to solve, by hand, a given linear program using the simplex method.<br />
#** Be able to solve, by computer, a given linear program.<br />
#** Explain and use the dual formulation of a given linear program.<br />
#* Markov Chains (6 lectures)<br />
#** Be able to describe a Markov chain.<br />
#** Identify when a Markov chain is regular, irregular, and absorbing.<br />
#** Describe how to determine the stable probabilities of a regular Markov chain.<br />
#** Describe how to compute the fundamental matrix of an absorbing Markov chain.<br />
#** Be able to apply ideas from Markov chains, and formulate real-world problems. Determine the transition matrix and states of a given application.<br />
<br />
</div><br />
<br />
<br />
<br />
=== Prerequisites ===<br />
Math 110<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
Finite Mathematics, 5th edition, Daniel P. Maki and Maynard Thompson, McGraw-Hill 2005<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|118]]</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_116:_Essentials_of_Calculus&diff=1827Math 116: Essentials of Calculus2012-05-10T20:02:45Z<p>Roundy: /* Offered */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Essentials of Calculus<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(1:1:0)<br />
<br />
=== Offered ===<br />
Fall (1st and 2nd block), Winter (1st and 2nd block), Spring, Summer<br />
<br />
=== Prerequisite ===<br />
Math 110<br />
<br />
=== Description ===<br />
This course gives a brief overview of differential calculus. Topics covered include limits, derivatives, and applications of differentiation to optimization of functions.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
# Review of Algebra (4 lectures)<br />
#* Determine the graph of a line given two points, a point and a slope, the general form of a line, the slope-intercept form of a line<br />
#* Determine the slope of a line given two coordinate points, the equation of a line given the slope and a point, equation of a line given two points, and the x- & y-intercept of a line<br />
#* Find the secant line of a function on an interval, and use that to understand the average rate of change of a function<br />
#* Approximate the tangent line at a point, and use that to understand the instantaneous rate of change of a function<br />
# Limits and Derivatives (4 lectures)<br />
#* Determine the limit of standard functions, e.g., polynomials, rational functions, exponentials, logarithms<br />
#* Determine the limits of more complicated functions composed of simpler functions.<br />
#* Define the derivative, take derivatives of polynomials using definition<br />
#* Derive the differentiation rules for polynomials, exponentials, logarithms<br />
# Product, Quotient, and Chain Rules (2 lectures)<br />
#* Derive the product, quotient, and chain rules<br />
#* Use the product, quotient, and chain rules to compute complicated derivatives composed of simpler functions<br />
# Optimization and Applications (4 lectures)<br />
#* State the derivative rules for local extreme<br />
#* Use the derivative rules to find the local extrema of a function on an interval. Then find the global maximum (or minimum) of a function on an interval<br />
#* Use the derivative to solve problems in business, e.g., maximize profits, minimize costs, etc.<br />
#* Use Newton's method for root finding to locate local extrema. <br />
<br />
</div><br />
<br />
<br />
=== Prerequisites ===<br />
Math 110<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<br />
=== Note: ===<br />
This course will be taught on the block schedule, 2 hours/week for 7 weeks. The grade of pass/fail will be completely determined by the final exam. Students will need to get 80% on the exam to pass. Exams can be retaken weekly in the testing center during the semester and daily during finals (there will be multiple versions of the exam available). The purpose of this class is for the students in the business school to understand the idea of a derivative and how to use it to optimize a function.<br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
=== Additional topics ===<br />
<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|116]]</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_499R:_Senior_Thesis&diff=1826Math 499R: Senior Thesis2012-05-03T21:25:04Z<p>Roundy: /* Title: Senior Thesis */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Senior Thesis<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(.5-3:0:3)<br />
<br />
=== Offered ===<br />
<br />
=== Prerequisite ===<br />
<br />
=== Description ===<br />
<br />
=== Note ===<br />
<br />
== Desired Learning Outcomes ==<br />
The student will complete a strong Senior Thesis which is compatible with the policies of the Honors Program and meets the standards set by the student's thesis advisor.<br />
<br />
=== Prerequisites ===<br />
Good standing in the BYU Honors program, and permission of the thesis advisor.<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
</div><br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|499]]</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_499R:_Senior_Thesis&diff=1825Math 499R: Senior Thesis2012-05-03T21:24:45Z<p>Roundy: /* (Credit Hours:Lecture Hours:Lab Hours) */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title: Senior Thesis ===<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(.5-3:0:3)<br />
<br />
=== Offered ===<br />
<br />
=== Prerequisite ===<br />
<br />
=== Description ===<br />
<br />
=== Note ===<br />
<br />
== Desired Learning Outcomes ==<br />
The student will complete a strong Senior Thesis which is compatible with the policies of the Honors Program and meets the standards set by the student's thesis advisor.<br />
<br />
=== Prerequisites ===<br />
Good standing in the BYU Honors program, and permission of the thesis advisor.<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
</div><br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|499]]</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_499R:_Senior_Thesis&diff=1824Math 499R: Senior Thesis2012-05-03T21:23:45Z<p>Roundy: /* Title: Senior Thesis */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title: Senior Thesis ===<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
<br />
=== Offered ===<br />
<br />
=== Prerequisite ===<br />
<br />
=== Description ===<br />
<br />
=== Note ===<br />
<br />
== Desired Learning Outcomes ==<br />
The student will complete a strong Senior Thesis which is compatible with the policies of the Honors Program and meets the standards set by the student's thesis advisor.<br />
<br />
=== Prerequisites ===<br />
Good standing in the BYU Honors program, and permission of the thesis advisor.<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
</div><br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|499]]</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_499R:_Senior_Thesis&diff=1823Math 499R: Senior Thesis2012-05-03T21:23:33Z<p>Roundy: /* Title */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title: Senior Thesis ===<br />
Senior Thesis<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
<br />
=== Offered ===<br />
<br />
=== Prerequisite ===<br />
<br />
=== Description ===<br />
<br />
=== Note ===<br />
<br />
== Desired Learning Outcomes ==<br />
The student will complete a strong Senior Thesis which is compatible with the policies of the Honors Program and meets the standards set by the student's thesis advisor.<br />
<br />
=== Prerequisites ===<br />
Good standing in the BYU Honors program, and permission of the thesis advisor.<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
</div><br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|499]]</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_499R:_Senior_Thesis&diff=1822Math 499R: Senior Thesis2012-05-03T21:23:04Z<p>Roundy: /* Title */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Senior Thesis<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
<br />
=== Offered ===<br />
<br />
=== Prerequisite ===<br />
<br />
=== Description ===<br />
<br />
=== Note ===<br />
<br />
== Desired Learning Outcomes ==<br />
The student will complete a strong Senior Thesis which is compatible with the policies of the Honors Program and meets the standards set by the student's thesis advisor.<br />
<br />
=== Prerequisites ===<br />
Good standing in the BYU Honors program, and permission of the thesis advisor.<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
</div><br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|499]]</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_313:_Elementary_Linear_Algebra&diff=1812Math 313: Elementary Linear Algebra2011-09-09T17:07:48Z<p>Roundy: /* Prerequisites */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Elementary Linear Algebra.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
F, W, Sp, Su<br />
<br />
=== Prerequisite ===<br />
[[Math 112]] or [[Math 119]]. Students are recommended to take [[Math 290]] before taking Math 313<br />
<br />
=== Description ===<br />
Linear systems, matrices, vectors and vector spaces, linear transformations, determinants, inner product spaces, eigenvalues, and eigenvectors.<br />
<br />
This course is aimed at majors in mathematics, the physical sciences, engineering, and other students interested in applications of mathematics to their disciplines. Linear algebra is used more than any other form of advanced mathematics in industry and science. A key idea is the mathematical modeling of a problem via systems of linear equations.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
<br />
[[Math 113]]. [[Math 290]] is encouraged.<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
Upon completion of this course, the successful student will be able to:<br />
<br />
# Use Gaussian elimination to do all of the following: solve a linear system with reduced row echelon form, solve a linear system with row echelon form and backward substitution, find the inverse of a given matrix, and find the determinant of a given matrix.<br />
# Demonstrate proficiency at matrix algebra. For matrix multiplication demonstrate understanding of the associative law, the reverse order law for inverses and transposes, and the failure of the commutative law and the cancellation law.<br />
# Use Cramer's rule to solve a linear system.<br />
# Use cofactors to find the inverse of a given matrix and the determinant of a given matrix.<br />
# Determine whether a set with a given notion of addition and scalar multiplication is a vector space. Here, and in relevant numbers below, be familiar with both finite and infinite dimensional examples.<br />
# Determine whether a given subset of a vector space is a subspace.<br />
# Determine whether a given set of vectors is linearly independent, spans, or is a basis.<br />
# Determine the dimension of a given vector space or of a given subspace.<br />
# Find bases for the null space, row space, and column space of a given matrix, and determine its rank.<br />
# Demonstrate understanding of the Rank-Nullity Theorem and its applications.<br />
# Given a description of a linear transformation, find its matrix representation relative to given bases.<br />
# Demonstrate understanding of the relationship between similarity and change of basis.<br />
# Determine whether a real valued function on a vector space is a norm.<br />
# Find the norm of a vector and the angle between two vectors in an inner product space.<br />
# Use the inner product to express a vector in an inner product space as a linear combination of an orthogonal set of vectors.<br />
# Find the orthogonal complement of a given subspace.<br />
# Demonstrate understanding of the relationship of the row space, column space, and nullspace of a matrix (and its transpose) via orthogonal complements.<br />
# Demonstrate understanding of the Cauchy-Schwartz inequality and its applications.<br />
# Determine whether a vector space with a (sesquilinear) form is an inner product space.<br />
# Use the Gram-Schmidt process to find an orthonormal basis of an inner product space. Be capable of doing this both in '''R'''<sup>n</sup> and in function spaces that are inner product spaces.<br />
# Use least squares to fit a line (''y'' = ''ax'' + ''b'') to a table of data, plot the line and data points, and explain the meaning of least squares in terms of orthogonal projection.<br />
# Use the idea of least squares to find orthogonal projections onto subspaces and for polynomial curve fitting.<br />
# Find (real and complex) eigenvalues and eigenvectors of 2 &times; 2 or 3 &times; 3 matrices.<br />
# Determine whether a given matrix is diagonalizable. If so, find a matrix that diagonalizes it via similarity.<br />
# Demonstrate understanding of the relationship between eigenvalues of a square matrix and its determinant, its trace, and its invertibility/singularity.<br />
# Identify symmetric matrices and orthogonal matrices.<br />
# Find a matrix that orthogonally diagonalizes a given symmetric matrix.<br />
# Know and be able to apply the spectral theorem for symmetric matrices.<br />
# Correctly define terms and give examples relating to the above concepts.<br />
# Prove basic theorems about the above concepts.<br />
# Prove or disprove statements relating to the above concepts.<br />
# Be adept at hand computation for row reduction, matrix inversion and similar problems; also, use MATLAB or a similar program for linear algebra problems.<br />
<br />
<br />
<br />
</div><br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Math 314]], [[Math 334]], [[Math 342]], [[Math 355]], [[Math 371]], [[Math 431]], [[Math 485]], [[Math 570]]. It is clear from this list that it is imperative to cover all the minimal learning outcomes.<br />
<br />
[[Category:Courses|313]]</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_435:_Mathematical_Finance&diff=1759Math 435: Mathematical Finance2011-05-09T21:07:21Z<p>Roundy: /* Prerequisite */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Mathematical Finance.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
W<br />
<br />
=== Prerequisite ===<br />
[[One of Math 431, Stat 341, Stat 370]].<br />
<br />
=== Description ===<br />
The binomial asset pricing model (discrete probability). Martingales, pricing of derivative securities, random walk in financial models, random interest rates.<br />
<br />
== Desired Learning Outcomes ==<br />
The minimal expectation for this course is that students learn about mathematical finance ''in the context of discrete time and finite state-spaces.'' It is therefore not required that students be taught about Brownian motion, the Black-Scholes model, etc.<br />
<br />
=== Prerequisites ===<br />
Students should have had an introductory course in probability.<br />
<br />
=== Minimal learning outcomes ===<br />
Within the context mentioned above, students should be able to compute prices for derivative securities. They should be conversant with the standard terminology of mathematical finance and be able to use this terminology correctly in answering questions. At a minimum, students should understand the following concepts in the context of binomial decision trees:<br />
<div style="-moz-column-count:2; column-count:2;"><br />
# Martingales<br />
# Markov processes<br />
# Arbitrage<br />
# Risk neutrality<br />
# State prices<br><br><br />
# Options<br />
#* Call and put<br />
#* American and European<br />
# Stopping times<br />
# Simple random walks<br />
# Interest rate models<br />
</div><br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
* Steven E. Shreve, ''Stochastic Calculus for Finance I: The Binomial Asset Pricing Model'', Springer, 2005.<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
None.<br />
<br />
[[Category:Courses|435]]</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_435:_Mathematical_Finance&diff=1758Math 435: Mathematical Finance2011-05-09T21:06:14Z<p>Roundy: /* Prerequisite */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Mathematical Finance.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
W<br />
<br />
=== Prerequisite ===<br />
[[One of Math 431, Stat 341, Stat 370.]].<br />
<br />
=== Description ===<br />
The binomial asset pricing model (discrete probability). Martingales, pricing of derivative securities, random walk in financial models, random interest rates.<br />
<br />
== Desired Learning Outcomes ==<br />
The minimal expectation for this course is that students learn about mathematical finance ''in the context of discrete time and finite state-spaces.'' It is therefore not required that students be taught about Brownian motion, the Black-Scholes model, etc.<br />
<br />
=== Prerequisites ===<br />
Students should have had an introductory course in probability.<br />
<br />
=== Minimal learning outcomes ===<br />
Within the context mentioned above, students should be able to compute prices for derivative securities. They should be conversant with the standard terminology of mathematical finance and be able to use this terminology correctly in answering questions. At a minimum, students should understand the following concepts in the context of binomial decision trees:<br />
<div style="-moz-column-count:2; column-count:2;"><br />
# Martingales<br />
# Markov processes<br />
# Arbitrage<br />
# Risk neutrality<br />
# State prices<br><br><br />
# Options<br />
#* Call and put<br />
#* American and European<br />
# Stopping times<br />
# Simple random walks<br />
# Interest rate models<br />
</div><br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
* Steven E. Shreve, ''Stochastic Calculus for Finance I: The Binomial Asset Pricing Model'', Springer, 2005.<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
None.<br />
<br />
[[Category:Courses|435]]</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_686R:_Topics_in_Algebraic_Number_Theory.&diff=1757Math 686R: Topics in Algebraic Number Theory.2011-05-04T03:46:41Z<p>Roundy: /* Minimal learning outcomes */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Topics in Algebraic Number Theory.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Prerequisite ===<br />
Math 372 and permission of the Instructor. In general, the prerequisites will depend on the material covered.<br />
<br />
=== Description ===<br />
Current topics of research interest.<br />
<br />
== Desired Learning Outcomes ==<br />
To gain familiarity with working in general settings, e.g. elliptic curves over number fields, and not just over rationals.<br />
=== Prerequisites ===<br />
Math 372 and permission of the Instructor. In general, the prerequisites will depend on the material covered.<br />
<br />
=== Minimal learning outcomes ===<br />
These cannot be specified uniquely for a topics course. The following example gives a clear indication of the level of difficulty appropriate to the course. Students should know the technical terms, and be able to implement the methods taught in the course to work associated problems, including proving results of suitable accessibility. <br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
#Generalization of the Unique Factorization Theorem from rationals to number fields; Basic definitions: <br />
#* Number fields<br />
#* Algebraic integers in a number field<br />
#* Integral bases<br />
#* Discriminant<br />
#* Norms of ideals<br />
#* Finiteness of ideals of bounded norm<br />
#* Class number<br />
#* Finiteness of class number<br />
#* Dedekind's Unique Factorization Theorem for ideals of a number field<br />
#Geometry of numbers: <br />
#* Minkowski's lemma on lattice points<br />
#* Logarithmic spaces<br />
#* Dirichlet's Unit Theorem for the units of the ring of integers of a number field<br />
#* Theorems of Minkowski and of Hermite on discriminants of number fields<br />
#Ramification Theory: <br />
#* Relative extensions<br />
#* Relative discriminant and Dedekind's criterion for ramification in terms of discriminant<br />
#* Higher ramification groups<br />
#* Hilbert theory of ramification<br />
#Splitting of Primes: <br />
#* Frobenius map<br />
#* Artin symbol<br />
#* Artin map<br />
#* Splitting of primes in Abelian extensions in terms of Artin map<br />
#* Rudimentary class field theory<br />
#* Examples - quadratic and cyclotomic extensions<br />
#Arithmetic of cyclotomic fields, and the Kronecker-Weber Theorem<br />
#Dedekind zeta function<br />
#* The class number formula - the formula which relates the residue of the Dedekind zeta function of a number field at s=1 to its class number, regulator and discriminant<br />
</div><br />
<br />
=== Textbooks ===<br />
<br />
Possible textbooks for this course include, (but are not limited to): <br />
#* D. Marcus, Number Fields (Universitext)<br />
<br />
<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|686]]</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_686R:_Topics_in_Algebraic_Number_Theory.&diff=1756Math 686R: Topics in Algebraic Number Theory.2011-05-04T03:46:14Z<p>Roundy: /* Minimal learning outcomes */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Topics in Algebraic Number Theory.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Prerequisite ===<br />
Math 372 and permission of the Instructor. In general, the prerequisites will depend on the material covered.<br />
<br />
=== Description ===<br />
Current topics of research interest.<br />
<br />
== Desired Learning Outcomes ==<br />
To gain familiarity with working in general settings, e.g. elliptic curves over number fields, and not just over rationals.<br />
=== Prerequisites ===<br />
Math 372 and permission of the Instructor. In general, the prerequisites will depend on the material covered.<br />
<br />
=== Minimal learning outcomes ===<br />
These cannot be specified uniquely for a topics course. The following example gives a clear indication of the level of difficulty appropriate to the course. Students should know the technical terms, and be able to implement the methods taught in the course to work associated problems, including proving results of suitable accessibility. <br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
#Generalization of the Unique Factorization Theorem from rationals to number fields; Basic definitions: <br />
#* Number fields<br />
#* Algebraic integers in a number field<br />
#* Integral bases<br />
#* Discriminant<br />
#* Norms of ideals<br />
#* Finiteness of ideals of bounded norm<br />
#* Class number<br />
#* Finiteness of class number<br />
#* Dedekind's Unique Factorization Theorem for ideals of a number field<br />
#Geometry of numbers: <br />
#* Minkowski's lemma on lattice points<br />
#* Logarithmic spaces<br />
#* Dirichlet's Unit Theorem for the units of the ring of integers of a number field<br />
#* Theorems of Minkowski and of Hermite on discriminants of number fields<br />
#Ramification Theory: <br />
#* Relative extensions<br />
#* Relative discriminant and Dedekind's criterion for ramification in terms of discriminant<br />
#* Higher ramification groups<br />
#* Hilbert theory of ramification<br />
#Splitting of Primes: <br />
#* Frobenius map<br />
#* Artin symbol<br />
#* Artin map<br />
#* Splitting of primes in Abelian extensions in terms of Artin map<br />
#* Rudimentary class field theory<br />
#* Examples - quadratic and cyclotomic extensions<br />
#Arithmetic of cyclotomic fields, and the Kronecker-Weber Theorem<br />
#Dedekind zeta function<br />
#* The class number formula - the formula which relates the residue of the Dedekind zeta function of a number field at s=1 to its class number, regulator and discriminant<br />
</div><br />
<br />
<br />
In addition, time permitting, the instructor may want to add to the list topics of special interest to him/her.<br />
<br />
=== Textbooks ===<br />
<br />
Possible textbooks for this course include, (but are not limited to): <br />
#* D. Marcus, Number Fields (Universitext)<br />
<br />
<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|686]]</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_352:_Introduction_to_Complex_Analysis&diff=1735Math 352: Introduction to Complex Analysis2011-02-05T00:16:51Z<p>Roundy: /* Prerequisite */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Introduction to Complex Analysis.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
F, W<br />
<br />
=== Prerequisite ===<br />
[[Math 290]], and either [[Math 341|Math 341]] or concurrent enrollment.<br />
<br />
=== Description ===<br />
Complex algebra, analytic functions, integration in the complex plane, infinite series, theory of residues, conformal mapping.<br />
<br />
== Desired Learning Outcomes ==<br />
This course is aimed at graduates majoring in mathematical and physical sciences and engineering. In addition to being an important branch of mathematics in its own right, complex analysis is an important tool for differential equations (ordinary and partial), algebraic geometry and number theory. Thus it is a core requirement for all mathematics majors. It contributes to all the expected learning outcomes of the Mathematics BS (see [http://learningoutcomes.byu.edu]).<br />
<br />
=== Prerequisites ===<br />
Students are expected to have completed and mastered [[Math 290]], and to have taken or to have concurrent enrollment in [[Math 341]] (Theory of Analysis) to provide the necessary understanding of the modes of thought of mathematical analysis.<br />
<br />
=== Minimal learning outcomes ===<br />
Students should achieve mastery of the topics listed below. This means that they should know all relevant definitions, the full statements of the major theorems, and examples of the various concepts. Further, students should be able to solve non-trivial problems related to these concepts, and prove simple theorems in analogy to proofs given by the instructor.<br />
<br />
# Complex numbers, moduli, exponential form, arguments of products and quotients, roots of complex numbers, regions in the complex plane.<br />
# Limits, including those involving the point at infinity. Open, closed and connected sets. Continuity, derivatives.<br />
# Analytic functions, Cauchy-Riemann equations, harmonic functions, finding the harmonic conjugate.<br />
# Elementary functions in the complex plane: exponential and log functions, complex exponents, trigonometric and hyperbolic functions and their inverses.<br />
# Contour integrals, upper bounds for moduli, primitives, Cauchy-Goursat theorem, Cauchy integral formulae, Liouville theorem, maximum modulus theorem.<br />
# Taylor series, Laurent series, integration and differentiation of power series, uniqueness of series representation, multiplication and division of power series.<br />
# Isolated singularities, behavior near a singularity. Residue theorem, its application to improper integrals, Jordan's lemma. Argument principle, Rouche's theorem.<br />
#Conformal mappings. Moebius transformations.<br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
These are at the instructor's discretion as time allows. Possibilities include applications of complex analysis in physics.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
This course is required for [[Math 532]] and [[Math 587|587]]. It is needed by anyone proceeding to graduate studies in mathematics. As a result it is essential that ALL required learning objectives be covered.<br />
<br />
[[Category:Courses|352]]</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_352:_Introduction_to_Complex_Analysis&diff=1734Math 352: Introduction to Complex Analysis2011-02-04T23:59:03Z<p>Roundy: /* Prerequisites */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Introduction to Complex Analysis.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
F, W<br />
<br />
=== Prerequisite ===<br />
[[Math 290]], and either [[Math 342|Math 342]] or concurrent enrollment.<br />
<br />
=== Description ===<br />
Complex algebra, analytic functions, integration in the complex plane, infinite series, theory of residues, conformal mapping.<br />
<br />
== Desired Learning Outcomes ==<br />
This course is aimed at graduates majoring in mathematical and physical sciences and engineering. In addition to being an important branch of mathematics in its own right, complex analysis is an important tool for differential equations (ordinary and partial), algebraic geometry and number theory. Thus it is a core requirement for all mathematics majors. It contributes to all the expected learning outcomes of the Mathematics BS (see [http://learningoutcomes.byu.edu]).<br />
<br />
=== Prerequisites ===<br />
Students are expected to have completed and mastered [[Math 290]], and to have taken or to have concurrent enrollment in [[Math 341]] (Theory of Analysis) to provide the necessary understanding of the modes of thought of mathematical analysis.<br />
<br />
=== Minimal learning outcomes ===<br />
Students should achieve mastery of the topics listed below. This means that they should know all relevant definitions, the full statements of the major theorems, and examples of the various concepts. Further, students should be able to solve non-trivial problems related to these concepts, and prove simple theorems in analogy to proofs given by the instructor.<br />
<br />
# Complex numbers, moduli, exponential form, arguments of products and quotients, roots of complex numbers, regions in the complex plane.<br />
# Limits, including those involving the point at infinity. Open, closed and connected sets. Continuity, derivatives.<br />
# Analytic functions, Cauchy-Riemann equations, harmonic functions, finding the harmonic conjugate.<br />
# Elementary functions in the complex plane: exponential and log functions, complex exponents, trigonometric and hyperbolic functions and their inverses.<br />
# Contour integrals, upper bounds for moduli, primitives, Cauchy-Goursat theorem, Cauchy integral formulae, Liouville theorem, maximum modulus theorem.<br />
# Taylor series, Laurent series, integration and differentiation of power series, uniqueness of series representation, multiplication and division of power series.<br />
# Isolated singularities, behavior near a singularity. Residue theorem, its application to improper integrals, Jordan's lemma. Argument principle, Rouche's theorem.<br />
#Conformal mappings. Moebius transformations.<br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
These are at the instructor's discretion as time allows. Possibilities include applications of complex analysis in physics.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
This course is required for [[Math 532]] and [[Math 587|587]]. It is needed by anyone proceeding to graduate studies in mathematics. As a result it is essential that ALL required learning objectives be covered.<br />
<br />
[[Category:Courses|352]]</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_352:_Introduction_to_Complex_Analysis&diff=1733Math 352: Introduction to Complex Analysis2011-02-04T23:57:21Z<p>Roundy: /* Prerequisite */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Introduction to Complex Analysis.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
F, W<br />
<br />
=== Prerequisite ===<br />
[[Math 290]], and either [[Math 342|Math 342]] or concurrent enrollment.<br />
<br />
=== Description ===<br />
Complex algebra, analytic functions, integration in the complex plane, infinite series, theory of residues, conformal mapping.<br />
<br />
== Desired Learning Outcomes ==<br />
This course is aimed at graduates majoring in mathematical and physical sciences and engineering. In addition to being an important branch of mathematics in its own right, complex analysis is an important tool for differential equations (ordinary and partial), algebraic geometry and number theory. Thus it is a core requirement for all mathematics majors. It contributes to all the expected learning outcomes of the Mathematics BS (see [http://learningoutcomes.byu.edu]).<br />
<br />
=== Prerequisites ===<br />
Students are expected to have completed and mastered [[Math 314]] (Calculus of Several Variables) OR [[Math 342]] (the second part of Theory of Analysis) to provide the necessary understanding of the modes of thought of mathematical analysis.<br />
<br />
=== Minimal learning outcomes ===<br />
Students should achieve mastery of the topics listed below. This means that they should know all relevant definitions, the full statements of the major theorems, and examples of the various concepts. Further, students should be able to solve non-trivial problems related to these concepts, and prove simple theorems in analogy to proofs given by the instructor.<br />
<br />
# Complex numbers, moduli, exponential form, arguments of products and quotients, roots of complex numbers, regions in the complex plane.<br />
# Limits, including those involving the point at infinity. Open, closed and connected sets. Continuity, derivatives.<br />
# Analytic functions, Cauchy-Riemann equations, harmonic functions, finding the harmonic conjugate.<br />
# Elementary functions in the complex plane: exponential and log functions, complex exponents, trigonometric and hyperbolic functions and their inverses.<br />
# Contour integrals, upper bounds for moduli, primitives, Cauchy-Goursat theorem, Cauchy integral formulae, Liouville theorem, maximum modulus theorem.<br />
# Taylor series, Laurent series, integration and differentiation of power series, uniqueness of series representation, multiplication and division of power series.<br />
# Isolated singularities, behavior near a singularity. Residue theorem, its application to improper integrals, Jordan's lemma. Argument principle, Rouche's theorem.<br />
#Conformal mappings. Moebius transformations.<br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
These are at the instructor's discretion as time allows. Possibilities include applications of complex analysis in physics.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
This course is required for [[Math 532]] and [[Math 587|587]]. It is needed by anyone proceeding to graduate studies in mathematics. As a result it is essential that ALL required learning objectives be covered.<br />
<br />
[[Category:Courses|352]]</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_352:_Introduction_to_Complex_Analysis&diff=1732Math 352: Introduction to Complex Analysis2011-02-04T23:57:01Z<p>Roundy: /* Prerequisite */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Introduction to Complex Analysis.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
F, W<br />
<br />
=== Prerequisite ===<br />
[[Math 290]], and either [[Math 342|342]] or concurrent enrollment.<br />
<br />
=== Description ===<br />
Complex algebra, analytic functions, integration in the complex plane, infinite series, theory of residues, conformal mapping.<br />
<br />
== Desired Learning Outcomes ==<br />
This course is aimed at graduates majoring in mathematical and physical sciences and engineering. In addition to being an important branch of mathematics in its own right, complex analysis is an important tool for differential equations (ordinary and partial), algebraic geometry and number theory. Thus it is a core requirement for all mathematics majors. It contributes to all the expected learning outcomes of the Mathematics BS (see [http://learningoutcomes.byu.edu]).<br />
<br />
=== Prerequisites ===<br />
Students are expected to have completed and mastered [[Math 314]] (Calculus of Several Variables) OR [[Math 342]] (the second part of Theory of Analysis) to provide the necessary understanding of the modes of thought of mathematical analysis.<br />
<br />
=== Minimal learning outcomes ===<br />
Students should achieve mastery of the topics listed below. This means that they should know all relevant definitions, the full statements of the major theorems, and examples of the various concepts. Further, students should be able to solve non-trivial problems related to these concepts, and prove simple theorems in analogy to proofs given by the instructor.<br />
<br />
# Complex numbers, moduli, exponential form, arguments of products and quotients, roots of complex numbers, regions in the complex plane.<br />
# Limits, including those involving the point at infinity. Open, closed and connected sets. Continuity, derivatives.<br />
# Analytic functions, Cauchy-Riemann equations, harmonic functions, finding the harmonic conjugate.<br />
# Elementary functions in the complex plane: exponential and log functions, complex exponents, trigonometric and hyperbolic functions and their inverses.<br />
# Contour integrals, upper bounds for moduli, primitives, Cauchy-Goursat theorem, Cauchy integral formulae, Liouville theorem, maximum modulus theorem.<br />
# Taylor series, Laurent series, integration and differentiation of power series, uniqueness of series representation, multiplication and division of power series.<br />
# Isolated singularities, behavior near a singularity. Residue theorem, its application to improper integrals, Jordan's lemma. Argument principle, Rouche's theorem.<br />
#Conformal mappings. Moebius transformations.<br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
These are at the instructor's discretion as time allows. Possibilities include applications of complex analysis in physics.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
This course is required for [[Math 532]] and [[Math 587|587]]. It is needed by anyone proceeding to graduate studies in mathematics. As a result it is essential that ALL required learning objectives be covered.<br />
<br />
[[Category:Courses|352]]</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_352:_Introduction_to_Complex_Analysis&diff=1731Math 352: Introduction to Complex Analysis2011-02-04T23:27:54Z<p>Roundy: /* Prerequisite */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Introduction to Complex Analysis.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
F, W<br />
<br />
=== Prerequisite ===<br />
[[Math 290]] or [[Math 342|342]] or concurrent enrollment.<br />
<br />
=== Description ===<br />
Complex algebra, analytic functions, integration in the complex plane, infinite series, theory of residues, conformal mapping.<br />
<br />
== Desired Learning Outcomes ==<br />
This course is aimed at graduates majoring in mathematical and physical sciences and engineering. In addition to being an important branch of mathematics in its own right, complex analysis is an important tool for differential equations (ordinary and partial), algebraic geometry and number theory. Thus it is a core requirement for all mathematics majors. It contributes to all the expected learning outcomes of the Mathematics BS (see [http://learningoutcomes.byu.edu]).<br />
<br />
=== Prerequisites ===<br />
Students are expected to have completed and mastered [[Math 314]] (Calculus of Several Variables) OR [[Math 342]] (the second part of Theory of Analysis) to provide the necessary understanding of the modes of thought of mathematical analysis.<br />
<br />
=== Minimal learning outcomes ===<br />
Students should achieve mastery of the topics listed below. This means that they should know all relevant definitions, the full statements of the major theorems, and examples of the various concepts. Further, students should be able to solve non-trivial problems related to these concepts, and prove simple theorems in analogy to proofs given by the instructor.<br />
<br />
# Complex numbers, moduli, exponential form, arguments of products and quotients, roots of complex numbers, regions in the complex plane.<br />
# Limits, including those involving the point at infinity. Open, closed and connected sets. Continuity, derivatives.<br />
# Analytic functions, Cauchy-Riemann equations, harmonic functions, finding the harmonic conjugate.<br />
# Elementary functions in the complex plane: exponential and log functions, complex exponents, trigonometric and hyperbolic functions and their inverses.<br />
# Contour integrals, upper bounds for moduli, primitives, Cauchy-Goursat theorem, Cauchy integral formulae, Liouville theorem, maximum modulus theorem.<br />
# Taylor series, Laurent series, integration and differentiation of power series, uniqueness of series representation, multiplication and division of power series.<br />
# Isolated singularities, behavior near a singularity. Residue theorem, its application to improper integrals, Jordan's lemma. Argument principle, Rouche's theorem.<br />
#Conformal mappings. Moebius transformations.<br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
These are at the instructor's discretion as time allows. Possibilities include applications of complex analysis in physics.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
This course is required for [[Math 532]] and [[Math 587|587]]. It is needed by anyone proceeding to graduate studies in mathematics. As a result it is essential that ALL required learning objectives be covered.<br />
<br />
[[Category:Courses|352]]</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_686R:_Topics_in_Algebraic_Number_Theory.&diff=1730Math 686R: Topics in Algebraic Number Theory.2011-01-29T01:10:26Z<p>Roundy: /* Prerequisites */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Topics in Algebraic Number Theory.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Prerequisite ===<br />
Math 372 and permission of the Instructor. In general, the prerequisites will depend on the material covered.<br />
<br />
=== Description ===<br />
Current topics of research interest.<br />
<br />
== Desired Learning Outcomes ==<br />
To gain familiarity with working in general settings, e.g. elliptic curves over number fields, and not just over rationals.<br />
=== Prerequisites ===<br />
Math 372 and permission of the Instructor. In general, the prerequisites will depend on the material covered.<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
#Generalization of the Unique Factorization Theorem from rationals to number fields; Basic definitions: <br />
#* Number fields<br />
#* Algebraic integers in a number field<br />
#* Integral bases<br />
#* Discriminant<br />
#* Norms of ideals<br />
#* Finiteness of ideals of bounded norm<br />
#* Class number<br />
#* Finiteness of class number<br />
#* Dedekind's Unique Factorization Theorem for ideals of a number field<br />
#Geometry of numbers: <br />
#* Minkowski's lemma on lattice points<br />
#* Logarithmic spaces<br />
#* Dirichlet's Unit Theorem for the units of the ring of integers of a number field<br />
#* Theorems of Minkowski and of Hermite on discriminants of number fields<br />
#Ramification Theory: <br />
#* Relative extensions<br />
#* Relative discriminant and Dedekind's criterion for ramification in terms of discriminant<br />
#* Higher ramification groups<br />
#* Hilbert theory of ramification<br />
#Splitting of Primes: <br />
#* Frobenius map<br />
#* Artin symbol<br />
#* Artin map<br />
#* Splitting of primes in Abelian extensions in terms of Artin map<br />
#* Rudimentary class field theory<br />
#* Examples - quadratic and cyclotomic extensions<br />
#Arithmetic of cyclotomic fields, and the Kronecker-Weber Theorem<br />
#Dedekind zeta function<br />
#* The class number formula - the formula which relates the residue of the Dedekind zeta function of a number field at s=1 to its class number, regulator and discriminant<br />
</div><br />
<br />
<br />
In addition, time permitting, the instructor may want to add to the list topics of special interest to him/her.<br />
<br />
=== Textbooks ===<br />
<br />
Possible textbooks for this course include, (but are not limited to): <br />
#* D. Marcus, Number Fields (Universitext)<br />
<br />
<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|686]]</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_686R:_Topics_in_Algebraic_Number_Theory.&diff=1729Math 686R: Topics in Algebraic Number Theory.2011-01-29T01:09:26Z<p>Roundy: /* Prerequisite */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Topics in Algebraic Number Theory.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Prerequisite ===<br />
Math 372 and permission of the Instructor. In general, the prerequisites will depend on the material covered.<br />
<br />
=== Description ===<br />
Current topics of research interest.<br />
<br />
== Desired Learning Outcomes ==<br />
To gain familiarity with working in general settings, e.g. elliptic curves over number fields, and not just over rationals.<br />
=== Prerequisites ===<br />
The pre-requisites in the Catalog are adequate. Depending on the instructor and with her/his permission, a sound understanding of basic concepts of complex analysis, abstract algebra and number theory might be adequate, perhaps at the level of Math 352, 371, 372 and 487.<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
#Generalization of the Unique Factorization Theorem from rationals to number fields; Basic definitions: <br />
#* Number fields<br />
#* Algebraic integers in a number field<br />
#* Integral bases<br />
#* Discriminant<br />
#* Norms of ideals<br />
#* Finiteness of ideals of bounded norm<br />
#* Class number<br />
#* Finiteness of class number<br />
#* Dedekind's Unique Factorization Theorem for ideals of a number field<br />
#Geometry of numbers: <br />
#* Minkowski's lemma on lattice points<br />
#* Logarithmic spaces<br />
#* Dirichlet's Unit Theorem for the units of the ring of integers of a number field<br />
#* Theorems of Minkowski and of Hermite on discriminants of number fields<br />
#Ramification Theory: <br />
#* Relative extensions<br />
#* Relative discriminant and Dedekind's criterion for ramification in terms of discriminant<br />
#* Higher ramification groups<br />
#* Hilbert theory of ramification<br />
#Splitting of Primes: <br />
#* Frobenius map<br />
#* Artin symbol<br />
#* Artin map<br />
#* Splitting of primes in Abelian extensions in terms of Artin map<br />
#* Rudimentary class field theory<br />
#* Examples - quadratic and cyclotomic extensions<br />
#Arithmetic of cyclotomic fields, and the Kronecker-Weber Theorem<br />
#Dedekind zeta function<br />
#* The class number formula - the formula which relates the residue of the Dedekind zeta function of a number field at s=1 to its class number, regulator and discriminant<br />
</div><br />
<br />
<br />
In addition, time permitting, the instructor may want to add to the list topics of special interest to him/her.<br />
<br />
=== Textbooks ===<br />
<br />
Possible textbooks for this course include, (but are not limited to): <br />
#* D. Marcus, Number Fields (Universitext)<br />
<br />
<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|686]]</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_116:_Essentials_of_Calculus&diff=1728Math 116: Essentials of Calculus2011-01-29T00:55:10Z<p>Roundy: /* Desired Learning Outcomes */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Essentials of Calculus<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(1:1:0)<br />
<br />
=== Offered ===<br />
Fall, Winter<br />
<br />
=== Prerequisite ===<br />
Math 110<br />
<br />
=== Description ===<br />
This course gives a brief overview of differential calculus. Topics covered include limits, derivatives, and applications of differentiation to optimization of functions.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
# Review of Algebra (4 lectures)<br />
#* Determine the graph of a line given two points, a point and a slope, the general form of a line, the slope-intercept form of a line<br />
#* Determine the slope of a line given two coordinate points, the equation of a line given the slope and a point, equation of a line given two points, and the x- & y-intercept of a line<br />
#* Find the secant line of a function on an interval, and use that to understand the average rate of change of a function<br />
#* Approximate the tangent line at a point, and use that to understand the instantaneous rate of change of a function<br />
# Limits and Derivatives (4 lectures)<br />
#* Determine the limit of standard functions, e.g., polynomials, rational functions, exponentials, logarithms<br />
#* Determine the limits of more complicated functions composed of simpler functions.<br />
#* Define the derivative, take derivatives of polynomials using definition<br />
#* Derive the differentiation rules for polynomials, exponentials, logarithms<br />
# Product, Quotient, and Chain Rules (2 lectures)<br />
#* Derive the product, quotient, and chain rules<br />
#* Use the product, quotient, and chain rules to compute complicated derivatives composed of simpler functions<br />
# Optimization and Applications (4 lectures)<br />
#* State the derivative rules for local extreme<br />
#* Use the derivative rules to find the local extrema of a function on an interval. Then find the global maximum (or minimum) of a function on an interval<br />
#* Use the derivative to solve problems in business, e.g., maximize profits, minimize costs, etc.<br />
#* Use Newton's method for root finding to locate local extrema. <br />
<br />
</div><br />
<br />
<br />
=== Prerequisites ===<br />
Math 110<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<br />
=== Note: ===<br />
This course will be taught on the block schedule, 2 hours/week for 7 weeks. The grade of pass/fail will be completely determined by the final exam. Students will need to get 80% on the exam to pass. Exams can be retaken weekly in the testing center during the semester and daily during finals (there will be multiple versions of the exam available). The purpose of this class is for the students in the business school to understand the idea of a derivative and how to use it to optimize a function.<br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
=== Additional topics ===<br />
<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|116]]</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_116:_Essentials_of_Calculus&diff=1727Math 116: Essentials of Calculus2011-01-29T00:54:40Z<p>Roundy: /* Desired Learning Outcomes */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Essentials of Calculus<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(1:1:0)<br />
<br />
=== Offered ===<br />
Fall, Winter<br />
<br />
=== Prerequisite ===<br />
Math 110<br />
<br />
=== Description ===<br />
This course gives a brief overview of differential calculus. Topics covered include limits, derivatives, and applications of differentiation to optimization of functions.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
# Review of Algebra (4 lectures)<br />
#* Determine the graph of a line given two points, a point and a slope, the general form of a line, the slope-intercept form of a line<br />
#* Determine the slope of a line given two coordinate points, the equation of a line given the slope and a point, equation of a line given two points, and the x- & y-intercept of a line<br />
#* Find the secant line of a function on an interval, and use that to understand the average rate of change of a function<br />
#* Approximate the tangent line at a point, and use that to understand the instantaneous rate of change of a function<br />
# Limits and Derivatives (4 lectures)<br />
#* Determine the limit of standard functions, e.g., polynomials, rational functions, exponentials, logarithms<br />
#* Determine the limits of more complicated functions composed of simpler functions.<br />
#* Define the derivative, take derivatives of polynomials using definition<br />
#* Derive the differentiation rules for polynomials, exponentials, logarithms<br />
# Product, Quotient, and Chain Rules (2 lectures)<br />
#* Derive the product, quotient, and chain rules<br />
#* Use the product, quotient, and chain rules to compute complicated derivatives composed of simpler functions<br />
# Optimization and Applications (4 lectures)<br />
#* State the derivative rules for local extreme<br />
#* Use the derivative rules to find the local extrema of a function on an interval. Then find the global maximum (or minimum) of a function on an interval<br />
#* Use the derivative to solve problems in business, e.g., maximize profits, minimize costs, etc.<br />
#* Use Newton's method for root finding to locate local extrema. <br />
<br />
</div><br />
<br />
<br />
=== Prerequisites ===<br />
Math 110<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<br />
=== Note: ===<br />
This course will be taught on the block schedule, 2 hours/week for 7 weeks. The grade of pass/fail will be completely determined by the final exam. Students will need to get 80% on the exam to pass. Exams can be retaken weekly in the testing center during the semester and daily during finals (there will be multiple versions of the exam available). The purpose of this class is for the students in the business school to understand the idea of a derivative and how to use it to optimize a function.<br />
<br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
=== Additional topics ===<br />
<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|116]]</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_116:_Essentials_of_Calculus&diff=1726Math 116: Essentials of Calculus2011-01-29T00:53:58Z<p>Roundy: /* Desired Learning Outcomes */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Essentials of Calculus<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(1:1:0)<br />
<br />
=== Offered ===<br />
Fall, Winter<br />
<br />
=== Prerequisite ===<br />
Math 110<br />
<br />
=== Description ===<br />
This course gives a brief overview of differential calculus. Topics covered include limits, derivatives, and applications of differentiation to optimization of functions.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
# Review of Algebra (4 lectures)<br />
#* Determine the graph of a line given two points, a point and a slope, the general form of a line, the slope-intercept form of a line<br />
#* Determine the slope of a line given two coordinate points, the equation of a line given the slope and a point, equation of a line given two points, and the x- & y-intercept of a line<br />
#* Find the secant line of a function on an interval, and use that to understand the average rate of change of a function<br />
#* Approximate the tangent line at a point, and use that to understand the instantaneous rate of change of a function<br />
# Limits and Derivatives (4 lectures)<br />
#* Determine the limit of standard functions, e.g., polynomials, rational functions, exponentials, logarithms<br />
#* Determine the limits of more complicated functions composed of simpler functions.<br />
#* Define the derivative, take derivatives of polynomials using definition<br />
#* Derive the differentiation rules for polynomials, exponentials, logarithms<br />
# Product, Quotient, and Chain Rules (2 lectures)<br />
#* Derive the product, quotient, and chain rules<br />
#* Use the product, quotient, and chain rules to compute complicated derivatives composed of simpler functions<br />
# Optimization and Applications (4 lectures)<br />
#* State the derivative rules for local extreme<br />
#* Use the derivative rules to find the local extrema of a function on an interval. Then find the global maximum (or minimum) of a function on an interval<br />
#* Use the derivative to solve problems in business, e.g., maximize profits, minimize costs, etc.<br />
#* Use Newton's method for root finding to locate local extrema. <br />
<br />
</div><br />
<br />
<br />
=== Prerequisites ===<br />
Math 110<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<br />
=== Comments ===<br />
Note: This course will be taught on the block schedule, 2 hours/week for 7 weeks. The grade of pass/fail will be completely determined by the final exam. Students will need to get 80% on the exam to pass. Exams can be retaken weekly in the testing center during the semester and daily during finals (there will be multiple versions of the exam available). The purpose of this class is for the students in the business school to understand the idea of a derivative and how to use it to optimize a function.<br />
<br />
<br />
<br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
=== Additional topics ===<br />
<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|116]]</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_116:_Essentials_of_Calculus&diff=1725Math 116: Essentials of Calculus2011-01-29T00:52:36Z<p>Roundy: /* Textbooks */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Essentials of Calculus<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(1:1:0)<br />
<br />
=== Offered ===<br />
Fall, Winter<br />
<br />
=== Prerequisite ===<br />
Math 110<br />
<br />
=== Description ===<br />
This course gives a brief overview of differential calculus. Topics covered include limits, derivatives, and applications of differentiation to optimization of functions.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
# Review of Algebra (4 lectures)<br />
#* Determine the graph of a line given two points, a point and a slope, the general form of a line, the slope-intercept form of a line<br />
#* Determine the slope of a line given two coordinate points, the equation of a line given the slope and a point, equation of a line given two points, and the x- & y-intercept of a line<br />
#* Find the secant line of a function on an interval, and use that to understand the average rate of change of a function<br />
#* Approximate the tangent line at a point, and use that to understand the instantaneous rate of change of a function<br />
# Limits and Derivatives (4 lectures)<br />
#* Determine the limit of standard functions, e.g., polynomials, rational functions, exponentials, logarithms<br />
#* Determine the limits of more complicated functions composed of simpler functions.<br />
#* Define the derivative, take derivatives of polynomials using definition<br />
#* Derive the differentiation rules for polynomials, exponentials, logarithms<br />
# Product, Quotient, and Chain Rules (2 lectures)<br />
#* Derive the product, quotient, and chain rules<br />
#* Use the product, quotient, and chain rules to compute complicated derivatives composed of simpler functions<br />
# Optimization and Applications (4 lectures)<br />
#* State the derivative rules for local extreme<br />
#* Use the derivative rules to find the local extrema of a function on an interval. Then find the global maximum (or minimum) of a function on an interval<br />
#* Use the derivative to solve problems in business, e.g., maximize profits, minimize costs, etc.<br />
#* Use Newton's method for root finding to locate local extrema. <br />
<br />
</div><br />
<br />
<br />
=== Prerequisites ===<br />
Math 110<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
=== Additional topics ===<br />
<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|116]]</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_116:_Essentials_of_Calculus&diff=1724Math 116: Essentials of Calculus2011-01-29T00:52:09Z<p>Roundy: /* Desired Learning Outcomes */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Essentials of Calculus<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(1:1:0)<br />
<br />
=== Offered ===<br />
Fall, Winter<br />
<br />
=== Prerequisite ===<br />
Math 110<br />
<br />
=== Description ===<br />
This course gives a brief overview of differential calculus. Topics covered include limits, derivatives, and applications of differentiation to optimization of functions.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
# Review of Algebra (4 lectures)<br />
#* Determine the graph of a line given two points, a point and a slope, the general form of a line, the slope-intercept form of a line<br />
#* Determine the slope of a line given two coordinate points, the equation of a line given the slope and a point, equation of a line given two points, and the x- & y-intercept of a line<br />
#* Find the secant line of a function on an interval, and use that to understand the average rate of change of a function<br />
#* Approximate the tangent line at a point, and use that to understand the instantaneous rate of change of a function<br />
# Limits and Derivatives (4 lectures)<br />
#* Determine the limit of standard functions, e.g., polynomials, rational functions, exponentials, logarithms<br />
#* Determine the limits of more complicated functions composed of simpler functions.<br />
#* Define the derivative, take derivatives of polynomials using definition<br />
#* Derive the differentiation rules for polynomials, exponentials, logarithms<br />
# Product, Quotient, and Chain Rules (2 lectures)<br />
#* Derive the product, quotient, and chain rules<br />
#* Use the product, quotient, and chain rules to compute complicated derivatives composed of simpler functions<br />
# Optimization and Applications (4 lectures)<br />
#* State the derivative rules for local extreme<br />
#* Use the derivative rules to find the local extrema of a function on an interval. Then find the global maximum (or minimum) of a function on an interval<br />
#* Use the derivative to solve problems in business, e.g., maximize profits, minimize costs, etc.<br />
#* Use Newton's method for root finding to locate local extrema. <br />
<br />
</div><br />
<br />
<br />
=== Prerequisites ===<br />
Math 110<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
<br />
=== Additional topics ===<br />
<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|116]]</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_116:_Essentials_of_Calculus&diff=1723Math 116: Essentials of Calculus2011-01-29T00:51:00Z<p>Roundy: /* Desired Learning Outcomes */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Essentials of Calculus<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(1:1:0)<br />
<br />
=== Offered ===<br />
Fall, Winter<br />
<br />
=== Prerequisite ===<br />
Math 110<br />
<br />
=== Description ===<br />
This course gives a brief overview of differential calculus. Topics covered include limits, derivatives, and applications of differentiation to optimization of functions.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
# Review of Algebra (4 lectures)<br />
# * Determine the graph of a line given two points, a point and a slope, the general form of a line, the slope-intercept form of a line<br />
# * Determine the slope of a line given two coordinate points, the equation of a line given the slope and a point, equation of a line given two points, and the x- & y-intercept of a line<br />
# * Find the secant line of a function on an interval, and use that to understand the average rate of change of a function<br />
# * Approximate the tangent line at a point, and use that to understand the instantaneous rate of change of a function<br />
# Limits and Derivatives (4 lectures)<br />
# * Determine the limit of standard functions, e.g., polynomials, rational functions, exponentials, logarithms<br />
# * Determine the limits of more complicated functions composed of simpler functions.<br />
# * Define the derivative, take derivatives of polynomials using definition<br />
# * Derive the differentiation rules for polynomials, exponentials, logarithms<br />
# Product, Quotient, and Chain Rules (2 lectures)<br />
# * Derive the product, quotient, and chain rules<br />
# * Use the product, quotient, and chain rules to compute complicated derivatives composed of simpler functions<br />
# Optimization and Applications (4 lectures)<br />
# * State the derivative rules for local extreme<br />
# * Use the derivative rules to find the local extrema of a function on an interval. Then find the global maximum (or minimum) of a function on an interval<br />
# * Use the derivative to solve problems in business, e.g., maximize profits, minimize costs, etc.<br />
# * Use Newton's method for root finding to locate local extrema. <br />
<br />
</div><br />
<br />
<br />
=== Prerequisites ===<br />
Math 110<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
<br />
=== Additional topics ===<br />
<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|116]]</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_116:_Essentials_of_Calculus&diff=1722Math 116: Essentials of Calculus2011-01-29T00:49:09Z<p>Roundy: /* Desired Learning Outcomes */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Essentials of Calculus<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(1:1:0)<br />
<br />
=== Offered ===<br />
Fall, Winter<br />
<br />
=== Prerequisite ===<br />
Math 110<br />
<br />
=== Description ===<br />
This course gives a brief overview of differential calculus. Topics covered include limits, derivatives, and applications of differentiation to optimization of functions.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
# Review of Algebra (4 lectures)<br />
# * Determine the graph of a line given two points, a point and a slope, the general form of a line, the slope-intercept form of a line<br />
# * Determine the slope of a line given two coordinate points, the equation of a line given the slope and a point, equation of a line given two points, and the x- & y-intercept of a line<br />
# * Find the secant line of a function on an interval, and use that to understand the average rate of change of a function<br />
# * Approximate the tangent line at a point, and use that to understand the instantaneous rate of change of a function<br />
# Limits and Derivatives (4 lectures)<br />
# * Determine the limit of standard functions, e.g., polynomials, rational functions, exponentials, logarithms<br />
# * Determine the limits of more complicated functions composed of simpler functions.<br />
# * Define the derivative, take derivatives of polynomials using definition<br />
# * Derive the differentiation rules for polynomials, exponentials, logarithms<br />
# Product, Quotient, and Chain Rules (2 lectures)<br />
# * Derive the product, quotient, and chain rules<br />
# * Use the product, quotient, and chain rules to compute complicated derivatives composed of simpler functions<br />
# Optimization and Applications (4 lectures)<br />
# * State the derivative rules for local extreme<br />
# * Use the derivative rules to find the local extrema of a function on an interval. Then find the global maximum (or minimum) of a function on an interval<br />
# * Use the derivative to solve problems in business, e.g., maximize profits, minimize costs, etc.<br />
# * Use Newton's method for root finding to locate local extrema. <br />
<br />
</div><br />
<br />
<br />
=== Prerequisites ===<br />
Math 110<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
<br />
=== Additional topics ===<br />
<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|116]]</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_116:_Essentials_of_Calculus&diff=1721Math 116: Essentials of Calculus2011-01-29T00:44:14Z<p>Roundy: /* Desired Learning Outcomes */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Essentials of Calculus<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(1:1:0)<br />
<br />
=== Offered ===<br />
Fall, Winter<br />
<br />
=== Prerequisite ===<br />
Math 110<br />
<br />
=== Description ===<br />
This course gives a brief overview of differential calculus. Topics covered include limits, derivatives, and applications of differentiation to optimization of functions.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
* Review of Algebra (4 lectures)<br />
o Determine the graph of a line given two points, a point<br />
<br />
<br />
=== Prerequisites ===<br />
Math 110<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
<br />
=== Additional topics ===<br />
<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|116]]</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_116:_Essentials_of_Calculus&diff=1720Math 116: Essentials of Calculus2011-01-29T00:42:20Z<p>Roundy: /* Prerequisites */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Essentials of Calculus<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(1:1:0)<br />
<br />
=== Offered ===<br />
Fall, Winter<br />
<br />
=== Prerequisite ===<br />
Math 110<br />
<br />
=== Description ===<br />
This course gives a brief overview of differential calculus. Topics covered include limits, derivatives, and applications of differentiation to optimization of functions.<br />
<br />
== Desired Learning Outcomes ==<br />
???<br />
=== Prerequisites ===<br />
Math 110<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
<br />
=== Additional topics ===<br />
<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|116]]</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_116:_Essentials_of_Calculus&diff=1719Math 116: Essentials of Calculus2011-01-29T00:41:53Z<p>Roundy: /* Description */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Essentials of Calculus<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(1:1:0)<br />
<br />
=== Offered ===<br />
Fall, Winter<br />
<br />
=== Prerequisite ===<br />
Math 110<br />
<br />
=== Description ===<br />
This course gives a brief overview of differential calculus. Topics covered include limits, derivatives, and applications of differentiation to optimization of functions.<br />
<br />
== Desired Learning Outcomes ==<br />
???<br />
=== Prerequisites ===<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
<br />
=== Additional topics ===<br />
<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|116]]</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_116:_Essentials_of_Calculus&diff=1718Math 116: Essentials of Calculus2011-01-29T00:41:28Z<p>Roundy: /* Prerequisite */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Essentials of Calculus<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(1:1:0)<br />
<br />
=== Offered ===<br />
Fall, Winter<br />
<br />
=== Prerequisite ===<br />
Math 110<br />
<br />
=== Description ===<br />
???<br />
<br />
== Desired Learning Outcomes ==<br />
???<br />
=== Prerequisites ===<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
<br />
=== Additional topics ===<br />
<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|116]]</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_116:_Essentials_of_Calculus&diff=1717Math 116: Essentials of Calculus2011-01-29T00:39:55Z<p>Roundy: /* Offered */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Essentials of Calculus<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(1:1:0)<br />
<br />
=== Offered ===<br />
Fall, Winter<br />
<br />
=== Prerequisite ===<br />
???<br />
<br />
=== Description ===<br />
???<br />
<br />
== Desired Learning Outcomes ==<br />
???<br />
=== Prerequisites ===<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
<br />
=== Additional topics ===<br />
<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|116]]</div>Roundyhttps://math.byu.edu/wiki/index.php?title=Math_118:_Finite_Mathematics&diff=1716Math 118: Finite Mathematics2011-01-29T00:39:14Z<p>Roundy: /* (Credit Hours:Lecture Hours:Lab Hours) */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Finite Mathematics<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
Fall, Winter<br />
<br />
=== Prerequisite ===<br />
Math 110<br />
<br />
=== Description ===<br />
Description: This course studies the basic elements and applications of finite mathematics. The first half of the class covers the language of set theory, principles of counting and combinatorics, probability theory for equally likely outcomes, elementary stochastic processes, conditional probabilities, and repeated experiments. The concept of a random variable is developed, along with expectation and variance. The second half of the class explores systems of linear equations and matrix algebra, linear programming, and Markov chains. This course considers a broad range of applications in business, the life sciences, and the social sciences.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
# Part 1: Probability Models (20 lectures, approx.)<br />
#* Sets Theory (3 lectures)<br />
#** Describe sets using set-builder notation.<br />
#** Solve problems involving set membership, subsets, intersections, unions, and complements of sets.<br />
#** Be able to identify the various partitions of a Venn diagram.<br />
#** Determine the number of elements in a partition, based on set counting rules for unions, complements, and products.<br />
#* Combinatorics & Counting (6 lectures)<br />
#** Describe the sample space of an experiment and the set of all possible outcomes using trees and the multiplicative principle, as appropriate.<br />
#** Explain what a permutation is, and how many permutations there are for a given set.<br />
#** Demonstrate how to count more sophisticated permutation problems involving products and restrictions to sets.<br />
#** Use the notion of partitions to reduce permutation problems to combinations, where the order does not matter.<br />
#** Be able to solve various hybrid counting problems with and without replacement, with and without order.<br />
#** Be able to apply ideas from combinatorics and counting, and formulate real-world problems.<br />
#* Probability (8 lectures)<br />
#** Be able to describe the notions of outcomes and events in probability, and the axioms of a probability space.<br />
#** Use ideas from combinatorics to determine the probabilities of various events, with equally likely outcomes.<br />
#** Demonstrate understanding of probabilistic independence.<br />
#** Describe a stochastic process, and be able to compute probabilities of events on trees.<br />
#** Explain Bayes rule and demonstrate proficiency with conditional probability.<br />
#** Be able to use Bayes formula to compute probabilities of various conditionals.<br />
#** Be proficient with Bernoulli trials, be able to solve basic problems.<br />
#** Be able to apply ideas from probability, and formulate real-world problems.<br />
#* Random Variables, Expected Values, Variance (3 lectures)<br />
#** Demonstrate understanding of what a random variable is. Describe the probability density function and the distribution of a given random variable.<br />
#** Show how to compute the expectation, variance, and standard deviation of a random variable.<br />
#** Be able to read a table for a normal distribution and compute the probabilities of given events.<br />
#** Be able to apply ideas of uncertainty, and formulate real-world problems.<br />
# Part 2: Linear Models (20 lectures, approx.)<br />
#* Systems of Linear Equations (3 lectures)<br />
#** Use elimination and substitution methods to solve linear systems of two or three variables.<br />
#** Reduce a linear system into row echelon form, then solve with back substitution.<br />
#** Solve a linear system by transforming it into reduced row echelon form.<br />
#** Identify whether a system of linear equations has no solution, exactly one solution, or infinitely many solutions.<br />
#* Matrix Algebra and Applications (3 lectures)<br />
#** Perform matrix algebra operations: addition, multiplication.<br />
#** Compute the inverse of a matrix, identify when a matrix is not invertible.<br />
#** Study in detail at least one application involving matrix algebra (e.g., Leontief economic models).<br />
#* Linear Programming (8 lectures)<br />
#** Formulate linear programming problems from various application areas, such as business, resource management, etc.<br />
#** Describe the constraints, the feasible set, and the objective function of a given linear optimization problem.<br />
#** Solve a linear program using the graphical method, when possible.<br />
#** Explain the standard form of a linear program.<br />
#** Describe the following concepts from the simplex method: slack variable, pivot column, tableau.<br />
#** Be able to solve, by hand, a given linear program using the simplex method.<br />
#** Be able to solve, by computer, a given linear program.<br />
#** Explain and use the dual formulation of a given linear program.<br />
#* Markov Chains (6 lectures)<br />
#** Be able to describe a Markov chain.<br />
#** Identify when a Markov chain is regular, irregular, and absorbing.<br />
#** Describe how to determine the stable probabilities of a regular Markov chain.<br />
#** Describe how to compute the fundamental matrix of an absorbing Markov chain.<br />
#** Be able to apply ideas from Markov chains, and formulate real-world problems. Determine the transition matrix and states of a given application.<br />
<br />
</div><br />
<br />
<br />
<br />
=== Prerequisites ===<br />
Math 110<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
Finite Mathematics, 5th edition, Daniel P. Maki and Maynard Thompson, McGraw-Hill 2005<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|118]]</div>Roundy